#7576506

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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1603 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)(10,14)(11,12)(17,18,22,26)(19,23,29,37)(20,24,32,40)(21,25,34,42)(27,35,31,39)(28,36,30,38)(33,41,45,48)(43,46,44,47);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19)(18,23)(20,31)(21,28)(22,29)(24,39)(25,36)(26,37)(27,32)(30,34)(33,44)(35,40)(38,42)(41,47)(43,45)(46,48);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20,22,32)(18,24,26,40)(19,27,29,31)(21,33,34,45)(23,35,37,39)(25,41,42,48)(28,43,30,44)(36,46,38,47);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,16,14,15)(19,28)(21,34)(23,36)(25,42)(27,43)(29,30)(31,44)(33,45)(35,46)(37,38)(39,47)(41,48);(2,8)(6,7)(9,12,13,11)(10,15,14,16)(19,29)(20,32)(23,37)(24,40)(28,30)(33,45)(36,38)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,e^c=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, 7483702);