#8093499

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	#52687	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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)(3,8)(4,9)(5,12)(6,7)(10,15)(11,14)(13,16)(17,18,23,26)(19,27,22,24)(20,21,31,25)(28,29,30,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(3,11)(4,13)(8,14)(9,16)(17,19,23,22)(18,24,26,27)(20,28,31,30)(21,32,25,29)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,7)(3,11)(9,16)(10,12)(17,20)(18,25)(19,28)(21,26)(22,30)(23,31)(24,29)(27,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,21,23,25)(18,20,26,31)(19,29,22,32)(24,30,27,28)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(33,35,39,38)(34,42,36,40)(37,43,46,45)(41,48,44,47);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,22,23,19)(18,24,26,27)(20,30,31,28)(21,32,25,29)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000695);