#6368728

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

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

Polycyclic presentation:

2^9|b^2=hi,c^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,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	#72306547, #72306692, #72306705
maximal normal subgroup of	#72306547, #72306692, #72306705

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275924);