#6368617

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368558
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

1516 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,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=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306657, #72306678, #72306697
maximal normal subgroup of	#72306657, #72306678, #72306697

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275813);