#7015613

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1284 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 6922809);