#8464167

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	#53519	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8371363);