#317143

order	512
nilpotency class	3
isoclinism	isoclinic class of #251818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2535
prime radical	C₂³	#525
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

474 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,f^b=fh,d^c=dhi,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224339);