#365076

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1636 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,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, 272272);