#6368802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368766
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

1531 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)(3,8)(4,9)(5,7)(6,11)(10,13)(12,15)(14,16)(18,24)(20,31)(21,22)(25,37)(26,32)(27,35)(28,43)(29,34)(38,42)(39,44)(40,41)(47,48);(1,3,4,11)(2,6,9,8)(5,12,13,16)(7,14,10,15)(17,18)(19,25)(20,32)(21,24)(22,35)(23,27)(26,33)(28,44)(29,40)(30,41)(31,38)(34,37)(36,42)(39,46)(43,47)(45,48);(2,7)(5,13)(6,14)(8,15)(9,10)(12,16)(17,19,23,30)(18,25,27,41)(20,28,31,43)(21,34,22,29)(24,37,35,40)(26,39,42,48)(32,44,38,47)(33,46,36,45);(1,3)(2,8)(4,11)(5,12)(6,9)(7,15)(10,14)(13,16)(17,20)(18,26)(19,28)(21,33)(22,36)(23,31)(24,38)(25,39)(27,42)(29,45)(30,43)(32,35)(34,46)(37,47)(40,44)(41,48);(3,11)(6,8)(12,16)(14,15)(17,21,23,22)(18,24,27,35)(19,29,30,34)(20,33,31,36)(25,40,41,37)(26,38,42,32)(28,45,43,46)(39,44,48,47)

Polycyclic presentation:

2^9|b^2=i,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,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

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306547, #72306677, #72306708
maximal normal subgroup of	#72306547, #72306677, #72306708

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275998);