#8093364

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000560);