#10320320

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	#44156	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,30)(28,29)(31,32)(33,34)(35,42)(36,41)(37,40)(38,43)(39,45)(44,47)(46,48);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(19,20)(24,26)(27,29)(28,30)(33,35)(34,39)(36,44)(37,46)(38,45)(40,47)(41,48)(42,43);(2,7)(3,10)(8,13)(11,16)(17,19,23,20)(18,24,25,26)(21,27,31,29)(22,28,32,30)(33,36,38,37)(34,40,43,41)(35,44,45,46)(39,47,42,48);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20,23,19)(18,24,25,26)(21,27,31,29)(22,30,32,28)(33,37,38,36)(34,41,43,40)(35,44,45,46)(39,47,42,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21,23,31)(18,22,25,32)(19,27,20,29)(24,28,26,30)(33,35)(34,42)(36,44)(37,46)(38,45)(39,43)(40,48)(41,47);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,15)(10,16)(17,22)(18,21)(19,28)(20,30)(23,32)(24,27)(25,31)(26,29)(33,34)(35,39)(36,40)(37,41)(38,43)(42,45)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227516);