#10324631

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

4134 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,f^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, 10231827);