#9936395

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

4182 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,24)(19,22,20,27)(21,25,30,32)(26,31,28,29)(33,34,39,40)(35,42,37,36)(38,46,47,41)(43,44,45,48);(4,11)(8,14)(10,15)(12,16)(17,19)(18,22)(20,23)(21,26)(24,27)(25,31)(28,30)(29,32)(33,35,39,37)(34,36,40,42)(38,43,47,45)(41,44,46,48);(2,7)(3,9)(8,14)(10,15)(18,24)(19,20)(25,32)(26,28)(34,40)(36,42)(41,46)(44,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,19)(18,22,24,27)(21,28,30,26)(25,31,32,29)(33,36)(34,35)(37,40)(38,44)(39,42)(41,43)(45,46)(47,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,25)(19,26)(20,28)(22,31)(23,30)(24,32)(27,29)(33,37,39,35)(34,36,40,42)(38,43,47,45)(41,48,46,44);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,19)(20,24)(21,29)(23,27)(25,28)(26,32)(30,31)(33,38,39,47)(34,41,40,46)(35,43,37,45)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9843591);