#10336793

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

4144 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)(3,8)(4,10)(5,11)(6,12)(7,9)(13,16)(14,15)(17,18,24,28)(19,29,31,25)(20,23,21,26)(22,27,32,30)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,3)(2,8)(4,5)(6,9)(7,12)(10,11)(13,14)(15,16)(17,19,24,31)(18,25,28,29)(20,30,21,27)(22,23,32,26);(2,9)(3,12)(5,13)(10,15)(17,20,24,21)(18,26,28,23)(19,30,31,27)(22,29,32,25)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,4)(2,10)(3,13)(5,12)(6,11)(7,14)(8,16)(9,15)(17,21,24,20)(18,23,28,26)(19,30,31,27)(22,29,32,25)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,5,7,13)(2,11,9,16)(3,4,12,14)(6,15,8,10)(17,22)(18,27)(19,23)(20,25)(21,29)(24,32)(26,31)(28,30)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,6)(2,3)(4,11)(5,15)(7,8)(9,12)(10,13)(14,16)(17,23)(18,20)(19,22)(21,28)(24,26)(25,27)(29,30)(31,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,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, 10243989);