#323830

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

678 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=egi,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,f^c=fh,g^c=gi

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, 231026);