#8458123

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

4177 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=h,e^2=hi,f^2=h,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, 8365319);