#8097799

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3888 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004995);