#8436829

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,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, 8344025);