#6907984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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	#37656	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1994 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,28,36)(20,30,32,45)(21,34,31,25)(24,38,39,33)(27,40,42,48)(29,44,41,37)(35,46,47,43);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,23,26,36)(20,27,32,42)(21,29,31,41)(24,35,39,47)(25,37,34,44)(30,40,45,48)(33,43,38,46);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,20,31,38,22,32,21,33)(18,24,34,45,26,39,25,30)(19,27,41,46,28,42,29,43)(23,35,44,48,36,47,37,40);(2,6)(5,7)(10,14)(13,15)(19,28)(20,32)(23,36)(24,39)(29,41)(30,45)(33,38)(37,44);(3,8)(5,7)(20,32)(24,39)(27,42)(30,45)(33,38)(35,47)(40,48)(43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6815180);