#8271306

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

4083 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=i,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8178502);