#439306

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=efg,c^2=ef,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346502);