#358106

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

1730 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)(5,6,10,14,11,16,9,15)(7,17,20,28,21,31,19,12)(8,22,26,30,24,32,25,13)(18,34,36,29,35,27,23,33);(5,7,9,19,11,21,10,20)(6,12,14,17,16,28,15,31)(8,23,25,35,24,36,26,18)(13,29,22,27,30,33,32,34);(2,4)(5,8,9,25,11,24,10,26)(6,13,14,22,16,30,15,32)(7,18,19,23,21,35,20,36)(12,27,17,33,28,34,31,29);(8,24)(13,30)(18,35)(22,32)(23,36)(25,26)(27,34)(29,33)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,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, 265302);