#299308

order	512
nilpotency class	3
isoclinism	isoclinic class of #299301
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60718	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206504);