#9782958

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

4100 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9690154);