#432998

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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,22,24,34,25,30,26,29)(23,31,35,28,33,27,32,36);(5,7,9,12)(6,10,8,11)(13,15,16,19)(14,17,18,20)(21,23,26,32,25,33,24,35)(22,27,29,28,30,31,34,36);(5,8,9,6)(7,11,12,10)(14,18)(17,20)(21,23)(22,28)(24,32)(25,33)(26,35)(27,29)(30,36)(31,34);(2,4)(7,12)(10,11)(21,24,25,26)(22,29,30,34)(23,32,33,35)(27,36,31,28)

Polycyclic presentation:

2^9|a^2=eg,b^2=efi,c^2=ei,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi,g^c=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, 340194);