#6857304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1381 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18)(19,28)(20,33)(21,27)(22,24)(23,38)(25,32)(26,36)(29,45)(30,46)(31,39)(34,44)(35,43)(37,42)(40,48)(41,47);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15)(18,24)(19,29)(26,33)(27,38)(30,34)(31,32)(42,43)(47,48);(9,10,13,12)(11,15,16,14)(17,19,21,31,22,29,23,32)(18,25,38,45,24,39,27,28)(20,34,37,47,36,30,35,48)(26,41,42,44,33,40,43,46);(2,6)(5,7)(17,20,22,36)(18,26,24,33)(19,30,29,34)(21,37,23,35)(25,40,39,41)(27,43,38,42)(28,44,45,46)(31,48,32,47);(3,8)(5,7)(9,12,13,10)(11,14,16,15)(17,21,22,23)(18,27,24,38)(19,31,29,32)(20,35,36,37)(25,28,39,45)(26,42,33,43)(30,47,34,48)(40,46,41,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6764500);