#6368948

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^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	#72306601, #72306632, #72306705
maximal normal subgroup of	#72306601, #72306632, #72306705

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276144);