#6821712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1307 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)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,29,31,24)(20,33,35,25)(21,37,32,27)(23,28,38,43)(30,40,47,45)(34,41,39,44)(36,48,46,42);(1,3)(2,5)(4,8)(6,7)(17,19,22,31)(18,24,26,29)(20,34,35,39)(21,38,32,23)(25,41,33,44)(27,43,37,28)(30,46,47,36)(40,48,45,42);(10,13)(14,16)(17,20,23,36,22,35,38,46)(18,25,28,42,26,33,43,48)(19,30,32,34,31,47,21,39)(24,40,37,41,29,45,27,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,26)(20,35)(24,29)(27,37)(28,43)(30,47)(34,39)(36,46);(3,8)(5,7)(17,21,38,31,22,32,23,19)(18,27,43,29,26,37,28,24)(20,34,46,30,35,39,36,47)(25,41,48,40,33,44,42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6728908);