#8533087

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	#2635	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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8440283);