#500848

order	512
nilpotency class	4
isoclinism	isoclinic class of #500848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1001 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)(7,18)(8,23)(9,20)(10,22)(11,17)(12,16)(13,15)(14,27)(19,34)(21,29)(24,26)(25,35)(28,32)(30,31)(33,36);(1,2)(3,4)(5,7,18,6,13,22,20,17,12,21,29,16,11,9,10,15)(8,24,33,31,27,25,34,32,28,19,35,14,30,36,26,23);(2,4)(5,8)(6,14)(7,19)(9,25)(10,26)(11,30)(12,28)(13,27)(15,32)(16,31)(17,23)(18,33)(20,34)(21,24)(22,36)(29,35);(5,9,18,15,13,7,20,6,12,22,29,17,11,21,10,16)(8,25,33,32,27,19,34,14,28,36,35,23,30,24,26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239838
maximal normal subgroup of	#72239838

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408044);