#6367291

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367198
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

1552 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)(3,14)(4,10)(5,12)(6,11)(7,13)(8,15)(9,16)(17,18,22,25)(19,27,30,38)(20,24,34,41)(21,35,36,26)(23,29,39,31)(28,43,32,46)(33,47,48,42)(37,44,40,45);(1,3,7,9)(2,8,13,4)(5,10,6,15)(11,16,12,14)(17,19,22,30)(18,23,25,39)(20,32,34,28)(21,29,36,31)(24,40,41,37)(26,38,35,27)(33,45,48,44)(42,43,47,46);(1,4)(2,9)(3,13)(5,16)(6,14)(7,8)(10,11)(12,15)(17,20,22,34)(18,24,25,41)(19,28,30,32)(21,33,36,48)(23,37,39,40)(26,42,35,47)(27,43,38,46)(29,44,31,45);(1,4)(2,10)(3,12)(5,14)(6,16)(7,8)(9,11)(13,15)(19,29)(21,36)(23,38)(26,35)(27,39)(28,44)(30,31)(32,45)(33,48)(37,46)(40,43)(42,47);(18,25)(19,30)(24,41)(26,35)(28,32)(29,31)(42,47)(44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284038, #72306794, #72306825, …
maximal normal subgroup of	#72284038, #72306794, #72306825, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274487);