#7569680

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	#36018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1745 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)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,28,36)(20,24,32,40)(21,25,33,41)(27,35,31,39)(29,37,44,47)(30,38,34,42)(43,46,45,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(20,30)(21,33)(24,38)(25,41)(27,43)(29,44)(31,45)(32,34)(35,46)(37,47)(39,48)(40,42);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,23)(20,31)(21,29)(22,28)(24,39)(25,37)(26,36)(27,32)(30,45)(33,44)(34,43)(35,40)(38,48)(41,47)(42,46);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,24)(19,27)(21,34)(22,32)(23,35)(25,42)(26,40)(28,31)(29,45)(30,33)(36,39)(37,48)(38,41)(43,44)(46,47);(2,6)(7,8)(10,14)(13,15)(19,28)(20,32)(23,36)(24,40)(29,44)(30,34)(37,47)(38,42)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^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,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, 7476876);