#8460725

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8367921);