#9947400

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	#53870	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,14)(12,16)(17,18)(19,22)(20,23)(21,26)(24,30)(25,28)(27,32)(29,31)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,46)(47,48);(2,7)(3,11)(9,16)(10,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32)(33,35,39,38)(34,40,43,44)(36,47,42,45)(37,48,41,46);(2,7)(3,11)(9,16)(10,13)(18,19)(23,25)(24,26)(31,32);(1,3,6,11)(2,8,7,14)(4,10,12,13)(5,9,15,16)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31)(33,36,39,42)(34,41,43,37)(35,45,38,47)(40,48,44,46);(1,4)(2,9)(3,10)(5,14)(6,12)(7,16)(8,15)(11,13)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,37,39,41)(34,42,43,36)(35,46,38,48)(40,47,44,45);(1,5,6,15)(2,10,7,13)(3,9,11,16)(4,8,12,14)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,38,39,35)(34,40,43,44)(36,47,42,45)(37,46,41,48)

Polycyclic presentation:

2^9|b^2=ghi,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, 9854596);