#9947402

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
prime radical	#42266	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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)(8,14)(9,12)(11,16)(13,15)(19,25)(22,26)(27,31)(30,32)(33,34,40,37)(35,44,39,41)(36,42,38,43)(45,48,46,47);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18,21,24)(19,26,25,22)(20,23,29,28)(27,32,31,30)(33,35,40,39)(34,41,37,44)(36,45,38,46)(42,47,43,48);(2,3)(7,9)(8,10)(15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,36,40,38)(34,42,37,43)(35,45,39,46)(41,47,44,48);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,37,40,34)(35,41,39,44)(36,43,38,42)(45,47,46,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,32)(30,31)(33,38,40,36)(34,43,37,42)(35,45,39,46)(41,47,44,48);(1,5)(2,8)(3,10)(4,11)(6,14)(7,15)(9,16)(12,13)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,39,40,35)(34,41,37,44)(36,46,38,45)(42,47,43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=gi,e^2=gh,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, 9854598);