#6507373

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25109	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 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)(3,6)(4,5)(7,8)(17,18,22,26)(19,27,28,40)(20,24,32,38)(21,34,29,25)(23,35,36,31)(30,41,42,48)(33,45,43,39)(37,46,47,44);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(19,28)(20,32)(23,36)(24,38)(27,40)(31,35)(33,43)(39,45);(9,11)(10,13)(12,16)(14,15)(17,19,29,35,22,28,21,31)(18,23,34,40,26,36,25,27)(20,30,43,46,32,42,33,44)(24,37,45,48,38,47,39,41);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,24)(19,30)(21,33)(22,32)(23,37)(25,39)(26,38)(27,41)(28,42)(29,43)(31,44)(34,45)(35,46)(36,47)(40,48);(10,14)(13,15)(19,28)(20,32)(23,36)(24,38)(27,40)(31,35)(33,43)(39,45)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6414569);