#7552519

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#45243	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1674 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)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,26)(19,23,30,37)(20,24,31,38)(21,25,33,40)(27,34,42,46)(28,35,29,36)(32,39,44,48)(41,45,43,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,28)(22,30)(24,34)(25,35)(26,37)(29,33)(31,42)(32,41)(36,40)(38,46)(39,45)(43,44)(47,48);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,22,31)(18,24,26,38)(19,27,30,42)(21,32,33,44)(23,34,37,46)(25,39,40,48)(28,41,29,43)(35,45,36,47);(9,11,12,16)(10,14,13,15)(19,28)(21,33)(23,35)(25,40)(27,41)(29,30)(32,44)(34,45)(36,37)(39,48)(42,43)(46,47);(1,4,5,3)(2,7,8,6)(20,31)(24,38)(27,42)(32,44)(34,46)(39,48)(41,43)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=g,e^2=f,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,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7459715);