#7094818

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

1308 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)(17,18,23,28)(19,29,21,26)(20,33,35,25)(22,38,30,27)(24,37,40,32)(31,41,34,43)(36,42,44,47)(39,45,46,48);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,30,37,23,21,22,32)(18,24,38,26,28,40,27,29)(20,31,44,48,35,34,36,45)(25,39,47,41,33,46,42,43);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,23,35)(18,25,28,33)(19,31,21,34)(22,36,30,44)(24,39,40,46)(26,41,29,43)(27,42,38,47)(32,45,37,48);(11,16)(13,15)(17,21,23,19)(18,26,28,29)(20,34,35,31)(22,32,30,37)(24,27,40,38)(25,41,33,43)(36,45,44,48)(39,42,46,47);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(20,35)(25,33)(31,34)(36,44)(39,46)(41,43)(42,47)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=gh,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,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, 7002014);