#10409993

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

4288 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10317189);