#499739

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
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

1322 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:

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

Polycyclic presentation:

2^9|c^2=i,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,e^b=eh,h^b=hi,f^c=fi,e^d=eh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250117
maximal normal subgroup of	#72250117

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406935);