#6367360

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367198
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,11)(4,8)(5,9)(6,13)(7,10)(12,15)(14,16)(17,18)(19,24)(20,32)(21,26)(22,35)(23,27)(25,33)(28,42)(29,38)(30,39)(31,45)(34,40)(36,41)(37,43)(44,47)(46,48);(1,3)(2,5)(4,9)(6,12)(7,14)(8,16)(10,15)(11,13)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,37)(26,38)(27,39)(32,42)(33,43)(34,44)(35,45)(36,46)(40,47)(41,48);(1,4,7,6)(2,8,10,13)(3,12,14,9)(5,11,15,16)(17,20,23,34)(18,25,27,41)(19,28,30,44)(21,33,22,36)(24,37,39,48)(26,40,35,32)(29,43,31,46)(38,47,45,42);(1,5,7,15)(2,9,10,12)(3,13,14,8)(4,11,6,16)(17,21,23,22)(18,26,27,35)(19,29,30,31)(20,33,34,36)(24,38,39,45)(25,40,41,32)(28,43,44,46)(37,47,48,42);(2,10)(3,14)(8,13)(9,12)(18,27)(19,30)(25,41)(26,35)(28,44)(29,31)(32,40)(43,46)

Polycyclic presentation:

2^9|c^2=hi,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306795, #72306796, #72306833
maximal normal subgroup of	#72306795, #72306796, #72306833

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274556);