#8492883

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3886 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=h,e^2=i,f^2=g,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, 8400079);