#376243

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₄×#121
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1304 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi

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, 283439);