#8093245

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=gh,e^2=h,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, 8000441);