#8095493

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	#52646	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002689);