#299458

order	512
nilpotency class	3
isoclinism	isoclinic class of #299405
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	#60817	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

995 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)(3,5)(4,11)(6,10)(7,14)(8,12)(9,15)(13,16)(17,18,24,19)(20,31,32,21)(22,30,23,27)(25,26,28,29);(1,3,5,2)(4,9,13,7)(6,10,12,8)(11,14,16,15)(17,19,22,30,24,18,23,27)(20,31,29,28,32,21,26,25);(1,4,12,15,5,13,6,14)(2,7,10,16,3,9,8,11)(17,20,23,26,24,32,22,29)(18,25,30,21,19,28,27,31);(2,3)(7,9)(8,10)(11,16)(17,21,22,25,24,31,23,28)(18,26,27,20,19,29,30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fi,d^2=ehi,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,d^b=di,g^b=gh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206654);