#9947265

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

4112 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)(4,7,12,9)(5,10,14,8)(11,16,13,15)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,34)(35,37)(36,44)(38,46)(39,40)(41,48)(42,43)(45,47);(2,3)(7,9)(8,10)(15,16)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,35)(34,37)(36,45)(38,43)(39,41)(40,48)(42,46)(44,47);(2,3)(7,9)(8,10)(15,16)(34,40)(35,41)(36,46)(43,47);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36)(34,38)(35,42)(37,47)(39,46)(40,44)(41,45)(43,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20,22,18)(19,23,27,24)(21,25,29,28)(26,31,32,30)(33,37)(34,35)(36,47)(38,42)(39,48)(40,41)(43,46)(44,45);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,38)(34,36)(35,43)(37,45)(39,44)(40,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=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, 9854461);