#8464245

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	#39909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=gh,e^2=gi,f^2=gi,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, 8371441);