#8096843

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,e^2=gh,f^2=h,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, 8004039);