#8741714

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3955 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8648910);