#8270577

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	#56571	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=gi,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8177773);