#9292992

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

3996 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=hi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9200188);