#265382

order	512
nilpotency class	3
isoclinism	isoclinic class of #265379
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,6,3)(4,12,14,16)(5,10,7,9)(8,15,11,13)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(1,3,5,10,6,2,7,9)(4,11,13,16,14,8,15,12)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,32)(28,31);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,32)(29,30);(1,4,7,15,6,14,5,13)(2,8,10,16,3,11,9,12)(20,27)(23,31)(25,32)(28,30)

Polycyclic presentation:

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

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, 172578);