#498814

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
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 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280369
maximal normal subgroup of	#72280369

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406010);