#10225401

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

4115 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)(3,9)(4,8)(5,10)(6,7)(11,14)(12,16)(13,15)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34,40,43)(35,39,38,41)(36,42,37,47)(44,46,45,48);(2,7)(3,10)(4,12)(14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,40,38)(34,41,43,39)(36,44,37,45)(42,48,47,46);(2,7)(3,10)(8,16)(11,13)(18,22)(23,25)(24,28)(30,31)(33,36,40,37)(34,42,43,47)(35,44,38,45)(39,46,41,48);(1,3)(2,5)(4,13)(6,10)(7,9)(8,14)(11,12)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,40,36)(34,42,43,47)(35,44,38,45)(39,48,41,46);(1,4)(2,8)(3,11)(5,15)(6,12)(7,16)(9,14)(10,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,38,40,35)(34,41,43,39)(36,45,37,44)(42,48,47,46);(1,5)(2,3)(4,14)(6,9)(7,10)(8,13)(11,16)(12,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,39,40,41)(34,38,43,35)(36,46,37,48)(42,45,47,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10132597);