#8810078

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

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, 8717274);