#6355436

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355407
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1597 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:

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

Polycyclic presentation:

2^9|b^2=f,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,d^c=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306857, #72306861, #72306925, …
maximal normal subgroup of	#72306857, #72306861, #72306925, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262632);