#441674

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fi,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,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, 348870);