#435617

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

811 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:

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

Polycyclic presentation:

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

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, 342813);