#358475

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1882 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^c=ei,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, 265671);