#8193882

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	#57588	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gi,e^2=i,f^2=gh,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, 8101078);