#365061

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
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	C₄×#121
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1626 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 272257);