#6368971

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

1499 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)(17,18,24,30)(19,31,36,42)(20,26,21,27)(22,29,23,28)(25,34,43,35)(32,41,38,48)(33,40,47,45)(37,46,39,44);(1,2)(3,11)(4,9)(5,7)(6,10)(8,12)(13,15)(14,16)(17,19,24,36)(18,25,30,43)(20,32,21,38)(22,35,23,34)(26,40,27,45)(28,31,29,42)(33,37,47,39)(41,44,48,46);(1,2)(3,8)(4,7)(5,9)(6,10)(11,12)(13,16)(14,15)(17,20,24,21)(18,26,30,27)(19,32,36,38)(22,39,23,37)(25,40,43,45)(28,46,29,44)(31,41,42,48)(33,34,47,35);(1,3,6,13)(2,8,10,16)(4,14,5,12)(7,15,9,11)(17,21,24,20)(18,27,30,26)(19,33,36,47)(22,39,23,37)(25,41,43,48)(28,46,29,44)(31,40,42,45)(32,34,38,35);(1,4,6,5)(2,9,10,7)(3,12,13,14)(8,15,16,11)(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,37,21,39)(25,42,43,31)(26,44,27,46)(32,33,38,47)(40,41,45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=h,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, #72306708
maximal normal subgroup of	#72306613, #72306708

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276167);