#358445

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

1890 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:

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

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,d^a=di,e^a=ei,f^a=fi,c^b=cg,f^b=fi,e^c=ei

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, 265641);