#9945513

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	#55451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=ghi,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, 9852709);