#9858595

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

4116 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=gi,f^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, 9765791);