#7566823

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1591 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)(9,10,12,14)(11,15,16,13)(17,18)(19,28)(20,32)(21,37)(22,27)(23,26)(24,31)(25,36)(29,39)(30,40)(33,42)(34,43)(35,44)(38,41)(45,48)(46,47);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,27,28)(20,33,36,29)(21,30,23,38)(25,42,32,39)(26,40,37,41)(34,45,35,46)(43,47,44,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,22,36)(18,25,27,32)(19,29,31,33)(21,34,23,35)(24,39,28,42)(26,43,37,44)(30,46,38,45)(40,48,41,47);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21)(18,26)(20,34)(22,23)(24,28)(25,43)(27,37)(30,38)(32,44)(35,36)(39,42)(45,46);(2,6)(7,8)(10,14)(13,15)(17,21,22,23)(18,26,27,37)(19,30,31,38)(20,35,36,34)(24,40,28,41)(25,44,32,43)(29,45,33,46)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fgi,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7474019);