#6915813

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1472 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)(17,18,22,23)(19,28,29,40)(20,32,33,25)(21,26,36,27)(24,38,37,31)(30,45,44,43)(34,42,35,41)(39,47,48,46);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(18,23)(19,29)(20,33)(26,27)(31,38)(34,35)(39,48)(43,45);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,21,31,22,29,36,38)(18,24,27,28,23,37,26,40)(20,30,35,47,33,44,34,46)(25,39,41,43,32,48,42,45);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(17,20,22,33)(18,25,23,32)(19,30,29,44)(21,35,36,34)(24,39,37,48)(26,42,27,41)(28,43,40,45)(31,47,38,46);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(17,21,22,36)(18,26,23,27)(19,31,29,38)(20,34,33,35)(24,40,37,28)(25,41,32,42)(30,46,44,47)(39,43,48,45)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgi,d^2=f,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,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6823009);