#6367232

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

1599 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:

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

Polycyclic presentation:

2^9|a^2=i,c^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	#72307011, #72307013, #72307019
maximal normal subgroup of	#72307011, #72307013, #72307019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274428);