#7582274

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	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1528 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,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, 7489470);