#10320385

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3857 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=gi,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, 10227581);