#7117790

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=gh,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7024986);