#10320394

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=ghi,e^2=g,f^2=h,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, 10227590);