#10360944

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

3955 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10268140);