#248811

order	512
nilpotency class	3
isoclinism	isoclinic class of #248743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#42962	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156007);