#452156

order	512
nilpotency class	3
isoclinism	isoclinic class of #417454
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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1359 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359352);