#7620921

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1262 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,27)(19,29,31,24)(20,33,36,25)(21,38,32,26)(23,28,39,44)(30,40,34,41)(35,42,45,46)(37,47,48,43);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,27,29)(20,34,36,30)(21,39,32,23)(25,41,33,40)(26,44,38,28)(35,37,45,48)(42,43,46,47);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,40)(26,42)(27,33)(28,43)(29,41)(31,34)(32,45)(38,46)(39,48)(44,47);(10,13)(14,16)(17,21,23,19,22,32,39,31)(18,26,28,24,27,38,44,29)(20,35,37,30,36,45,48,34)(25,42,43,40,33,46,47,41);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(18,27)(20,36)(24,29)(26,38)(28,44)(30,34)(35,45)(37,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=ghi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7528117);