#8431688

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	#43278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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)(3,11,6,9)(4,10,5,14)(12,16,13,15)(20,28)(24,31)(26,32)(29,30)(33,34,40,38)(35,45,46,41)(36,43,37,42)(39,44,47,48);(1,3,8,6)(2,9,7,11)(4,12,5,13)(10,15,14,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,35,40,46)(34,41,38,45)(36,39,37,47)(42,44,43,48);(1,4,8,5)(2,10,7,14)(3,12,6,13)(9,15,11,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,36,40,37)(34,42,38,43)(35,39,46,47)(41,44,45,48);(1,5,8,4)(2,10,7,14)(3,13,6,12)(9,15,11,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37,40,36)(34,43,38,42)(35,39,46,47)(41,44,45,48);(1,6,8,3)(2,11,7,9)(4,13,5,12)(10,16,14,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,40,34)(35,45,46,41)(36,43,37,42)(39,48,47,44);(1,7,8,2)(3,11,6,9)(4,14,5,10)(12,16,13,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,39)(34,44)(35,37)(36,46)(38,48)(40,47)(41,43)(42,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,f^2=gh,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, 8338884);