#10325553

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

4318 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)(3,10)(4,8)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(2,7)(3,8)(5,14)(12,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(2,7)(4,10)(9,13)(12,16)(18,20)(23,25)(24,28)(30,31)(37,45)(40,44)(42,48)(46,47);(1,3,6,8)(2,4,7,10)(5,11,14,15)(9,12,13,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,36,38,34)(35,39,43,41)(37,40,45,44)(42,47,48,46);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,35)(34,41)(36,39)(37,42)(38,43)(40,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,e^2=gh,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, 10232749);