#8436688

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	#52656	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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)(3,12,14,9)(4,10,5,15)(6,16,13,11)(17,18,24,22)(19,29,31,25)(20,27,21,26)(23,28,32,30)(33,34)(35,40)(36,47)(37,41)(38,42)(39,43)(44,45)(46,48);(1,3,8,14)(2,9,7,12)(4,13,5,6)(10,16,15,11)(17,19,24,31)(18,25,22,29)(20,30,21,28)(23,27,32,26)(33,35,39,46)(34,40,43,48)(36,37,42,44)(38,45,47,41);(1,4,8,5)(2,10,7,15)(3,13,14,6)(9,16,12,11)(17,20,24,21)(18,26,22,27)(19,30,31,28)(23,29,32,25)(33,36,39,42)(34,38,43,47)(35,44,46,37)(40,41,48,45);(1,5,8,4)(2,10,7,15)(3,6,14,13)(9,16,12,11)(17,21,24,20)(18,27,22,26)(19,30,31,28)(23,29,32,25)(33,35)(34,40)(36,44)(37,42)(38,41)(39,46)(43,48)(45,47);(1,6)(2,11)(3,4)(5,14)(7,16)(8,13)(9,10)(12,15)(17,22,24,18)(19,29,31,25)(20,27,21,26)(23,30,32,28)(33,37,39,44)(34,41,43,45)(35,42,46,36)(38,48,47,40);(1,7,8,2)(3,12,14,9)(4,15,5,10)(6,16,13,11)(17,23,24,32)(18,28,22,30)(19,27,31,26)(20,29,21,25)(33,38)(34,42)(35,45)(36,43)(37,48)(39,47)(40,44)(41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=g,e^2=h,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, 8343884);