#8095766

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

4049 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002962);