#441199

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35528	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1643 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egi,c^2=e,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,f^c=fi,h^d=hi

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, 348395);