#7524606

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1434 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18,23,29)(19,25,32,42)(20,33,31,26)(21,35,38,27)(22,40,24,28)(30,47,48,41)(34,43,36,44)(37,45,39,46);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,24)(21,34)(22,31)(23,32)(26,40)(27,44)(28,33)(29,42)(30,39)(35,43)(36,38)(37,48)(41,45)(46,47);(1,4,5,3)(2,6,7,8)(10,14)(11,15)(17,20,23,31)(18,26,29,33)(19,22,32,24)(21,36,38,34)(25,28,42,40)(27,44,35,43)(30,37,48,39)(41,45,47,46);(10,14)(13,16)(17,21,23,38)(18,27,29,35)(19,30,32,48)(20,34,31,36)(22,39,24,37)(25,41,42,47)(26,43,33,44)(28,46,40,45);(2,7)(6,8)(17,22,23,24)(18,28,29,40)(19,31,32,20)(21,37,38,39)(25,33,42,26)(27,45,35,46)(30,34,48,36)(41,43,47,44)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7431802);