#10102258

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

4141 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10009454);