#6993369

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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

1561 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)(17,18,23,24,22,27,19,28)(20,31,29,25,34,38,32,40)(21,26,36,37,35,41,30,42)(33,44,43,39,46,47,45,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,30,35,36)(25,38,40,31)(26,37,41,42)(33,45,46,43)(39,47,48,44);(3,8)(5,7)(9,11,14,10)(12,15,13,16)(17,20,22,34)(18,25,27,40)(19,29,23,32)(21,33,35,46)(24,31,28,38)(26,39,41,48)(30,43,36,45)(37,44,42,47);(2,6)(5,7)(9,12,14,13)(10,15,11,16)(17,21,22,35)(18,26,27,41)(19,30,23,36)(20,33,34,46)(24,37,28,42)(25,39,40,48)(29,43,32,45)(31,44,38,47);(9,13,14,12)(10,15,11,16)(21,35)(26,41)(30,36)(33,46)(37,42)(39,48)(43,45)(44,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fhi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6900565);