#8107243

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	#52363	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

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, 8014439);