#8456938

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=h,e^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, 8364134);