#8508921

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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)(4,14)(5,13)(6,11)(7,8)(10,15)(12,16)(17,18)(19,28)(20,26)(21,25)(22,24)(23,27)(29,31)(30,32)(33,34,39,40)(35,36,43,48)(37,41,38,42)(44,46,45,47);(2,8)(5,15)(9,14)(11,12)(17,19,23,22)(18,24,27,28)(20,29,21,30)(25,31,26,32)(34,40)(35,43)(41,42)(44,45);(1,3,7,4)(2,9,8,14)(5,12,15,11)(6,13,16,10)(17,20,23,21)(18,25,27,26)(19,29,22,30)(24,31,28,32)(33,35)(34,36)(37,44)(38,45)(39,43)(40,48)(41,46)(42,47);(1,4,7,3)(2,9,8,14)(5,11,15,12)(6,13,16,10)(17,21,23,20)(18,26,27,25)(19,29,22,30)(24,31,28,32)(33,36)(34,35)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);(1,5)(2,10)(3,12)(4,11)(6,9)(7,15)(8,13)(14,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31)(33,37,39,38)(34,41,40,42)(35,44,43,45)(36,46,48,47);(1,6)(2,11)(3,13)(4,10)(5,9)(7,16)(8,12)(14,15)(17,22,23,19)(18,24,27,28)(20,30,21,29)(25,31,26,32)(33,38,39,37)(34,42,40,41)(35,45,43,44)(36,47,48,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=gi,e^2=i,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, 8416117);