#8458672

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

4192 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)(3,7)(4,14)(5,9)(6,10)(8,11)(12,16)(13,15)(20,28)(24,31)(27,32)(29,30)(33,34,39,42)(35,38,41,36)(37,45,46,40)(43,44,48,47);(1,3,8,6)(2,7,11,10)(4,13,5,12)(9,16,14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,39,41)(34,36,42,38)(37,43,46,48)(40,44,45,47);(1,4,8,5)(2,9,11,14)(3,12,6,13)(7,15,10,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36,39,38)(34,35,42,41)(37,44,46,47)(40,43,45,48);(1,5,8,4)(2,9,11,14)(3,13,6,12)(7,15,10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,37)(34,40)(35,43)(36,44)(38,47)(39,46)(41,48)(42,45);(1,6,8,3)(2,10,11,7)(4,13,5,12)(9,16,14,15)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,7,8,10)(2,3,11,6)(4,15,5,16)(9,12,14,13)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,38,39,36)(34,41,42,35)(37,44,46,47)(40,43,45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,e^2=i,f^2=gi,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, 8365868);