#9494740

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

4193 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=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, 9401936);