#8180509

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087705);