#372919

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1158 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=i,d^2=fg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280115);