#7524350

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

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,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, 7431546);