#358586

order	512
nilpotency class	3
isoclinism	isoclinic class of #23414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1586 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265782);