#6367292

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

1526 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)(3,13)(4,14)(5,9)(6,11)(7,10)(8,12)(15,16)(17,18,22,25)(19,27,28,43)(20,24,33,41)(21,34,37,26)(23,35,38,30)(29,45,46,40)(31,48,47,39)(32,44,36,42);(1,3)(2,4)(5,12)(6,9)(7,14)(8,15)(10,16)(11,13)(17,19)(18,23)(20,31)(21,35)(22,28)(24,40)(25,38)(26,43)(27,34)(29,32)(30,37)(33,47)(36,46)(39,42)(41,45)(44,48);(1,4,8,5)(2,3,12,15)(6,13,7,16)(9,10,14,11)(19,28)(23,38)(27,43)(29,46)(30,35)(31,47)(39,48)(40,45);(1,5,8,4)(2,9,12,14)(3,10,15,11)(6,13,7,16)(17,20,22,33)(18,24,25,41)(19,29,28,46)(21,36,37,32)(23,39,38,48)(26,44,34,42)(27,45,43,40)(30,47,35,31);(18,25)(19,28)(24,41)(26,34)(29,46)(30,35)(31,47)(42,44)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=h,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	#72283802, #72306994, #72307025
maximal normal subgroup of	#72283802, #72306994, #72307025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274488);