#8096257

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	#45912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4135 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)(3,10,5,8)(4,13,15,6)(11,12,14,16)(17,18,24,28)(19,23,25,20)(21,29,32,26)(22,27,30,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,3,7,5)(2,8,9,10)(4,11,15,14)(6,12,13,16)(17,19,24,25)(18,20,28,23)(21,30,32,22)(26,31,29,27)(36,44)(39,46)(43,47)(45,48);(17,20,24,23)(18,25,28,19)(21,27,32,31)(22,26,30,29)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,45,46,48);(1,4)(2,6)(3,11)(5,14)(7,15)(8,12)(9,13)(10,16)(17,21)(18,26)(19,22)(20,27)(23,31)(24,32)(25,30)(28,29)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,46)(42,48);(1,5,7,3)(2,8,9,10)(4,14,15,11)(6,12,13,16)(17,22)(18,27)(19,21)(20,26)(23,29)(24,30)(25,32)(28,31)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48);(1,6)(2,4)(3,12)(5,16)(7,13)(8,11)(9,15)(10,14)(17,23,24,20)(18,25,28,19)(21,31,32,27)(22,26,30,29)(33,37)(34,40)(35,41)(36,45)(38,42)(39,47)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=gi,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, 8003453);