#8202491

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(3,9,10,8)(4,13,14,5)(11,12,16,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,7)(3,10)(4,14)(12,15)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(35,43)(36,46)(39,48)(40,45);(2,7)(5,13)(8,9)(12,15)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,35)(34,39)(36,44)(37,40)(38,43)(41,46)(42,48)(45,47);(1,3,6,10)(2,8,7,9)(4,11,14,16)(5,12,13,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31);(1,4)(2,5)(3,11)(6,14)(7,13)(8,12)(9,15)(10,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,5)(2,4)(3,12)(6,13)(7,14)(8,11)(9,16)(10,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=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, 8109687);