#10368621

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
prime radical	#55328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=hi,e^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, 10275817);