#371084

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

892 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)(5,6,10,15)(7,16,20,26)(8,21,22,12)(9,13,17,14)(11,24,27,19)(18,31,33,36)(23,29,32,30)(25,35,28,34);(1,3,4,2)(5,7,17,24,10,20,9,19)(6,11,13,26,15,27,14,16)(8,18,32,35,22,33,23,34)(12,28,30,31,21,25,29,36);(5,8,9,23,10,22,17,32)(6,12,14,29,15,21,13,30)(7,18,19,34,20,33,24,35)(11,25,16,31,27,28,26,36);(6,13,15,14)(8,22)(11,26,27,16)(12,29,21,30)(18,33)(23,32)(25,31,28,36)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278280);