#10444921

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

3958 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)(3,8,4,11)(5,9,13,14)(10,15,12,16)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32)(33,34)(35,39)(36,41)(37,47)(38,43)(40,46)(42,44)(45,48);(2,7)(8,11)(9,14)(15,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,32)(29,30)(33,35,38,45)(34,39,43,48)(36,44,46,37)(40,47,41,42);(1,3,6,4)(2,8,7,11)(5,10,13,12)(9,15,14,16)(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,36)(34,40)(35,44)(37,45)(38,46)(39,47)(41,43)(42,48);(1,4,6,3)(2,8,7,11)(5,10,13,12)(9,16,14,15)(17,20,23,18)(19,26,21,24)(22,25,30,28)(27,31,29,32)(33,36)(34,41)(35,44)(37,45)(38,46)(39,42)(40,43)(47,48);(1,2)(3,8)(4,11)(5,9)(6,7)(10,15)(12,16)(13,14)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,34)(35,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,37,38,44)(34,42,43,47)(35,36,45,46)(39,40,48,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=gi,e^2=i,f^2=h,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, 10352117);