#8187007

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#42555	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094203);