#8436922

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,e^2=gh,f^2=g,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, 8344118);