#8093360

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,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, 8000556);