#7618458

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1383 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)(10,14)(11,12)(17,18,23,28)(19,30,32,24)(20,25,36,43)(21,26,33,45)(22,29,38,27)(31,47,48,40)(34,41,39,44)(35,46,37,42);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,32)(18,24,28,30)(20,31,36,48)(21,38,33,22)(25,40,43,47)(26,29,45,27)(34,37,39,35)(41,46,44,42);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,31)(21,39)(22,37)(23,36)(24,40)(26,44)(27,46)(28,43)(29,42)(30,47)(32,48)(33,34)(35,38)(41,45);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,16,14,15)(17,21,22,19,23,33,38,32)(18,26,29,30,28,45,27,24)(20,34,37,48,36,39,35,31)(25,41,42,40,43,44,46,47);(2,6)(7,8)(9,12,13,11)(10,15,14,16)(17,22,23,38)(18,27,28,29)(19,21,32,33)(20,35,36,37)(24,26,30,45)(25,42,43,46)(31,34,48,39)(40,41,47,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=fhi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7525654);