#7589567

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(17,18,21,25)(19,27,30,23)(20,24,32,38)(22,26,34,40)(28,41,43,35)(29,37,31,36)(33,39,45,48)(42,47,44,46);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30)(18,23,25,27)(20,28,32,43)(22,29,34,31)(24,35,38,41)(26,36,40,37)(33,42,45,44)(39,46,48,47);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,21,32)(18,24,25,38)(19,28,30,43)(22,33,34,45)(23,35,27,41)(26,39,40,48)(29,42,31,44)(36,46,37,47);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(19,29)(22,34)(23,36)(26,40)(27,37)(28,42)(30,31)(33,45)(35,46)(39,48)(41,47)(43,44);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(20,32)(24,38)(28,43)(33,45)(35,41)(39,48)(42,44)(46,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fgi,e^2=fgi,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, 7496763);