#6985958

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1435 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)(9,10,12,13)(11,15,16,14)(17,18)(19,27)(20,31)(21,35)(22,28)(23,24)(25,32)(26,38)(29,42)(30,40)(33,48)(34,39)(36,47)(37,44)(41,45)(43,46);(1,3,7,4)(2,5,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,28,27)(20,32,34,29)(21,30,38,37)(25,31,42,39)(26,40,35,44)(33,46,36,45)(41,47,43,48);(1,4,7,3)(2,5,8,6)(10,13)(14,15)(17,20,19,29,22,34,23,32)(18,25,24,39,28,42,27,31)(21,36,30,46,38,33,37,45)(26,43,40,47,35,41,44,48);(1,5,7,6)(2,3,8,4)(9,11,12,16)(10,14,13,15)(17,21)(18,26)(19,30)(20,33)(22,38)(23,37)(24,40)(25,41)(27,44)(28,35)(29,45)(31,47)(32,46)(34,36)(39,48)(42,43);(1,6,7,5)(2,3,8,4)(17,19,22,23)(18,27,28,24)(20,29,34,32)(21,37,38,30)(25,31,42,39)(26,40,35,44)(33,46,36,45)(41,48,43,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=fhi,c^2=fg,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,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, 6893154);