#8097798

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	#52656	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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)(17,18,23,26)(19,27,21,24)(20,22,25,32)(28,31,30,29)(33,34,39,38)(35,41,36,40)(37,45,46,42)(43,47,44,48);(1,2)(3,8)(4,7)(5,9)(6,11)(10,15)(12,14)(13,16)(17,19,23,21)(18,24,26,27)(20,28,25,30)(22,29,32,31)(33,35,39,36)(34,40,38,41)(37,44,46,43)(42,48,45,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,23,25)(18,22,26,32)(19,28,21,30)(24,29,27,31)(33,36,39,35)(34,41,38,40)(37,44,46,43)(42,48,45,47);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,21,23,19)(18,24,26,27)(20,30,25,28)(22,29,32,31);(1,2)(3,6)(4,7)(5,9)(8,11)(10,13)(12,14)(15,16)(17,22)(18,25)(19,29)(20,26)(21,31)(23,32)(24,30)(27,28)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=gh,f^2=h,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, 8004994);