#6907830

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#30704	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1349 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)(4,8)(5,7)(17,18,23,25,21,27,22,28)(19,29,36,40,31,24,32,38)(20,26,37,41,34,44,35,43)(30,42,46,48,33,39,45,47);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,33)(21,31)(22,32)(23,36)(25,38)(26,42)(27,29)(28,40)(30,34)(35,46)(37,45)(39,44)(41,48)(43,47);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(18,25)(20,34)(22,23)(24,38)(26,43)(27,28)(29,40)(30,33)(32,36)(39,48)(41,44)(42,47);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,20,21,34)(18,26,27,44)(19,30,31,33)(22,35,23,37)(24,39,29,42)(25,41,28,43)(32,45,36,46)(38,47,40,48);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(18,27)(19,31)(20,34)(25,28)(32,36)(35,37)(39,42)(47,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6815026);