#343128

order	512
nilpotency class	3
isoclinism	isoclinic class of #343112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

605 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,14,7,13,6,12)(3,15,19,9,18,28,17,27)(4,20,26,29,23,10,22,30)(5,24,21,11,16,32,25,31)(33,34)(35,44,48,55)(36,41)(37,42)(38,43)(39,46,40,47)(45,57,64,62)(49,56,63,54)(50,58)(51,59)(52,60)(53,61);(1,3,7,18)(2,9,13,27)(4,21,23,25)(5,22,16,26)(6,19,8,17)(10,24,20,32)(11,30,31,29)(12,28,14,15)(33,35,38,48)(34,39,43,40)(36,49,51,63)(37,47,53,46)(41,57,59,62)(42,55,61,44)(45,52,64,50)(54,60,56,58);(1,4)(2,10)(3,16)(5,18)(6,22)(7,23)(8,26)(9,11)(12,29)(13,20)(14,30)(15,24)(17,21)(19,25)(27,31)(28,32)(33,35)(34,40)(36,45)(37,46)(38,48)(39,43)(41,56)(42,55)(44,61)(47,53)(49,50)(51,64)(52,63)(54,59)(57,60)(58,62);(1,5,7,16)(2,11,13,31)(3,4,18,23)(6,21,8,25)(9,20,27,10)(12,32,14,24)(15,29,28,30)(17,26,19,22)(33,36,38,51)(34,41,43,59)(35,45,48,64)(37,52,53,50)(39,54,40,56)(42,60,61,58)(44,62,55,57)(46,63,47,49)

Polycyclic presentation:

2^9|a^2=efi,b^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250324);