#8445752

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

4011 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)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(33,34,38,39)(35,36,43,40)(37,41,46,47)(42,44,48,45);(1,3,8,14)(2,9,7,12)(4,13,5,6)(10,16,15,11)(19,29)(20,31)(24,28)(25,30)(34,39)(36,40)(41,47)(44,45);(1,4,8,5)(2,10,7,15)(3,13,14,6)(9,16,12,11)(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,35)(34,40)(36,39)(37,42)(38,43)(41,45)(44,47)(46,48);(1,5,8,4)(2,10,7,15)(3,6,14,13)(9,16,12,11)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,36,38,40)(34,35,39,43)(37,45,46,44)(41,48,47,42);(1,6)(2,11)(3,4)(5,14)(7,16)(8,13)(9,10)(12,15)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25);(1,7,8,2)(3,12,14,9)(4,15,5,10)(6,16,13,11)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,37,38,46)(34,41,39,47)(35,42,43,48)(36,44,40,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=g,e^2=i,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, 8352948);