#8456000

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	#56299	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4217 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,e^2=gi,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, 8363196);