#8188509

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,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, 8095705);