#6368671

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

1536 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,e^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	#72306547, #72306688, #72306700
maximal normal subgroup of	#72306547, #72306688, #72306700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275867);