#8508931

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3892 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

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, 8416127);