#434888

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

1032 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)(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,19,20,17)(21,22)(23,30)(24,32)(25,28)(26,35)(27,31)(29,34)(33,36);(5,7,9,12)(6,10,8,11)(21,23,25,31)(22,27,28,30)(24,26,33,34)(29,36,35,32);(13,15,16,20)(14,17,18,19)(21,24,26,31,25,33,34,23)(22,27,29,32,28,30,35,36);(2,4)(5,8,9,6)(7,10,12,11)(22,28)(23,31)(24,33)(29,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=e,c^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,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, 342084);