#6368666

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

1497 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)(3,8)(4,9)(5,7)(6,10)(11,16)(12,14)(13,15)(17,18,23,28)(19,24,31,41)(20,26,33,42)(21,27,22,36)(25,35,43,32)(29,38,46,48)(30,39,34,40)(37,44,47,45);(2,7)(4,14)(5,16)(8,15)(9,13)(10,11)(17,19,23,31)(18,24,28,41)(20,29,33,46)(21,34,22,30)(25,37,43,47)(26,38,42,48)(27,40,36,39)(32,45,35,44);(1,3)(2,8)(4,11)(5,13)(6,12)(7,15)(9,16)(10,14)(17,20)(18,25)(19,29)(21,32)(22,35)(23,33)(24,37)(26,36)(27,42)(28,43)(30,44)(31,46)(34,45)(38,39)(40,48)(41,47);(1,4,6,15)(2,9,10,13)(3,11,12,7)(5,8,16,14)(17,20)(18,26)(19,29)(21,35)(22,32)(23,33)(24,38)(25,36)(27,43)(28,42)(30,45)(31,46)(34,44)(37,39)(40,47)(41,48);(2,10)(3,12)(7,11)(9,13)(17,21,23,22)(18,27,28,36)(19,30,31,34)(20,32,33,35)(24,39,41,40)(25,42,43,26)(29,44,46,45)(37,38,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=i,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	#72306654, #72306681, #72306697
maximal normal subgroup of	#72306654, #72306681, #72306697

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275862);