#8932082

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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)(3,8,11,15)(4,9,13,12)(5,10,16,14)(17,18,23,28)(19,27,22,24)(20,21,26,25)(29,32,31,30)(34,39)(37,45)(40,46)(44,48);(2,7)(4,13)(8,15)(10,14)(17,19,23,22)(18,24,28,27)(20,31,26,29)(21,32,25,30)(36,43)(40,46)(42,47)(44,48);(3,11)(5,16)(8,15)(10,14)(17,20,23,26)(18,25,28,21)(19,29,22,31)(24,32,27,30)(33,34,38,39)(35,37,41,45)(36,40,43,46)(42,44,47,48);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,32)(30,31)(33,35,38,41)(34,37,39,45)(36,42,43,47)(40,44,46,48);(1,4)(2,9)(3,10)(5,15)(6,13)(7,12)(8,16)(11,14)(17,21,23,25)(18,26,28,20)(19,30,22,32)(24,31,27,29)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,22,23,19)(18,27,28,24)(20,31,26,29)(21,32,25,30)(33,37)(34,41)(35,39)(36,44)(38,45)(40,47)(42,46)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839278);