#8096891

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

4229 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:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004087);