#358266

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
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:

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

Polycyclic presentation:

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

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, 265462);