#7582434

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1513 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)(20,33)(21,27)(22,28)(23,29)(25,32)(26,37)(30,42)(31,41)(34,48)(35,40)(36,46)(38,45)(39,43)(44,47);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,28,25)(20,34,37,47)(21,31,23,39)(26,44,33,48)(27,41,29,43)(30,36,35,38)(40,45,42,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,25)(20,35)(21,39)(22,32)(23,31)(24,28)(26,40)(27,41)(29,43)(30,37)(33,42)(34,36)(38,47)(44,46)(45,48);(10,14)(13,15)(17,20,22,37)(18,26,28,33)(19,30,32,35)(21,38,23,36)(24,40,25,42)(27,46,29,45)(31,47,39,34)(41,48,43,44);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,31,32,39)(20,36,37,38)(24,41,25,43)(26,45,33,46)(30,34,35,47)(40,44,42,48)

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=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7489630);