#7581899

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	#25575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1546 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)(17,18,21,23)(19,24,28,35)(20,30,32,25)(22,26,33,36)(27,41,43,37)(29,38,44,46)(31,39,34,40)(42,47,45,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(18,23)(20,31)(22,33)(24,35)(25,39)(27,42)(29,44)(30,40)(32,34)(37,47)(41,48)(43,45);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,21,28)(18,24,23,35)(20,27,32,43)(22,29,33,44)(25,37,30,41)(26,38,36,46)(31,42,34,45)(39,47,40,48);(10,14)(13,15)(17,20,21,32)(18,25,23,30)(19,27,28,43)(22,34,33,31)(24,37,35,41)(26,39,36,40)(29,45,44,42)(38,47,46,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(18,23)(19,28)(25,30)(26,36)(27,43)(29,44)(39,40)(42,45)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,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, 7489095);