#441676

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=f,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, 348872);