#9913252

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

4377 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820448);