#8425580

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

4091 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332776);