#245218

order	512
nilpotency class	3
isoclinism	isoclinic class of #245204
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1029 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 148 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)(3,7)(4,8)(5,9)(6,10)(11,17)(12,18)(13,19)(14,27)(15,22)(16,23)(20,25)(21,28)(24,31)(26,30)(29,32)(33,34)(35,39)(36,46)(37,49)(38,42)(40,47)(41,52)(43,60)(44,54)(45,58)(48,59)(50,55)(51,57)(53,62)(56,63)(61,64);(1,3,6,13)(2,7,10,19)(4,11,15,25)(5,12,16,26)(8,20,22,17)(9,18,23,30)(14,24,28,32)(21,31,27,29)(33,35)(34,39)(36,43)(37,50)(38,44)(40,56)(41,58)(42,54)(45,52)(46,60)(47,63)(48,61)(49,55)(51,62)(53,57)(59,64);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,21)(10,22)(12,24)(13,25)(16,28)(18,29)(19,20)(23,27)(26,32)(30,31)(33,36,38,48)(34,40,42,57)(35,43,44,61)(37,51,52,47)(39,53,54,56)(41,59,49,46)(45,63,50,62)(55,64,58,60);(1,5,6,16)(2,9,10,23)(3,12,13,26)(4,14,15,28)(7,18,19,30)(8,21,22,27)(11,24,25,32)(17,29,20,31)(35,44)(39,54)(43,61)(45,50)(53,56)(55,58)(60,64)(62,63)

Polycyclic presentation:

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

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, 152414);