#8463484

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=g,e^2=gi,f^2=ghi,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, 8370680);