#8757050

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	#55347	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664246);