#408155

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=ei,d^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315351);