#442812

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1056 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efg,c^2=e,d^2=efi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 350008);