#6368760

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368694
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

1593 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306666, #72306698, #72306700
maximal normal subgroup of	#72306666, #72306698, #72306700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275956);