#8095846

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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)(3,8)(4,7)(5,9)(6,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,29)(21,31)(23,26)(24,27)(25,32)(28,30)(34,39)(40,46)(41,47)(42,48);(4,12)(7,14)(10,16)(13,15)(17,19)(18,20)(21,23)(22,27)(24,29)(25,26)(28,32)(30,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,24,22)(18,19,27,29)(21,30,32,26)(23,25,28,31)(33,35,38,44)(34,40,39,46)(36,37,43,45)(41,42,47,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(17,21)(18,25)(19,28)(20,30)(22,26)(23,29)(24,32)(27,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,22,24,20)(18,19,27,29)(21,26,32,30)(23,25,28,31)(33,36)(34,41)(35,37)(38,43)(39,47)(40,42)(44,45)(46,48);(1,2)(3,6)(4,7)(5,9)(8,11)(10,13)(12,14)(15,16)(17,23,24,28)(18,26,27,30)(19,21,29,32)(20,25,22,31)(33,37)(34,42)(35,43)(36,44)(38,45)(39,48)(40,47)(41,46)

Polycyclic presentation:

2^9|c^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003042);