#371153

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1136 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,g^c=gi,f^d=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, 278349);