#10325234

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

4192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10232430);