#9706356

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9613552);