#6883876

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	#38032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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)(3,7)(4,6)(5,8)(17,18,23,27)(19,24,22,37)(20,30,33,47)(21,26,36,43)(25,28,40,32)(29,38,35,42)(31,46,34,48)(39,44,41,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,29,36,35)(25,30,40,47)(26,38,43,42)(31,44,34,45)(39,46,41,48);(2,6)(3,8)(10,14)(15,16)(17,20,22,32,23,33,19,28)(18,25,24,30,27,40,37,47)(21,34,35,44,36,31,29,45)(26,41,38,48,43,39,42,46);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,36)(18,26,27,43)(19,29,22,35)(20,31,33,34)(24,38,37,42)(25,39,40,41)(28,44,32,45)(30,46,47,48);(9,12,13,11)(10,15,14,16)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,35,36,29)(25,30,40,47)(26,42,43,38)(31,45,34,44)(39,48,41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fhi,e^2=fhi,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, 6791072);