#7094854

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1315 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ghi,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi

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, 7002050);