#6774974

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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)(17,18,22,27)(19,23,30,38)(20,25,32,41)(21,24,34,26)(28,36,31,40)(29,35,44,37)(33,39,46,42)(43,47,45,48);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,27,38)(20,31,32,28)(21,29,34,44)(24,35,26,37)(25,40,41,36)(33,45,46,43)(39,48,42,47);(10,13)(14,16)(18,24)(21,34)(23,35)(25,39)(26,27)(29,44)(33,46)(36,47)(37,38)(40,48)(41,42)(43,45);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20,22,32)(18,25,27,41)(19,28,30,31)(21,33,34,46)(23,36,38,40)(24,39,26,42)(29,43,44,45)(35,47,37,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(20,32)(25,41)(28,31)(33,46)(36,40)(39,42)(43,45)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6682170);