#465839

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373035);