#6368968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368846
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

1482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306613, #72306654, #72306657, …
maximal normal subgroup of	#72306613, #72306654, #72306657, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276164);