#7569168

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#36015	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(3,6)(4,5)(7,8)(11,12)(14,16)(17,18,21,25)(19,27,30,23)(20,24,32,40)(22,26,34,42)(28,45,47,37)(29,46,48,38)(31,43,35,39)(33,41,36,44);(2,5)(6,8)(9,10,13,15)(11,16,12,14)(17,19)(18,23)(20,28)(21,30)(22,35)(24,37)(25,27)(26,43)(29,33)(31,34)(32,47)(36,48)(38,41)(39,42)(40,45)(44,46);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,15,16)(17,20,21,32)(18,24,25,40)(19,28,30,47)(22,33,34,36)(23,37,27,45)(26,41,42,44)(29,31,48,35)(38,39,46,43);(9,12,13,11)(10,14,15,16)(17,20)(18,24)(19,29)(21,32)(22,36)(23,38)(25,40)(26,44)(27,46)(28,35)(30,48)(31,47)(33,34)(37,43)(39,45)(41,42);(2,5)(6,8)(18,25)(20,32)(23,27)(26,42)(28,47)(29,48)(33,36)(39,43)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=g,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,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7476364);