#8772493

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

3926 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)(3,10)(4,7)(5,8)(6,12)(9,15)(11,13)(14,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,32)(26,30)(28,31)(33,34,38,41)(35,37,40,46)(36,39,45,48)(42,44,43,47);(1,3,7,8)(2,5,4,10)(6,14,15,11)(9,16,12,13)(18,22)(23,30)(24,27)(31,32)(35,40)(37,46)(42,43)(44,47);(2,4)(5,10)(9,12)(13,16)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31)(33,35,38,40)(34,37,41,46)(36,43,45,42)(39,47,48,44);(1,4,7,2)(3,10,8,5)(6,12,15,9)(11,16,14,13)(17,20,21,29)(18,24,22,27)(19,25,26,28)(23,31,30,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,43)(41,48)(46,47);(1,5,7,10)(2,8,4,3)(6,13,15,16)(9,14,12,11)(17,19)(18,23)(20,25)(21,26)(22,30)(24,31)(27,32)(28,29)(33,36)(34,39)(35,42)(37,44)(38,45)(40,43)(41,48)(46,47);(1,6,7,15)(2,9,4,12)(3,11,8,14)(5,13,10,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)(33,37)(34,40)(35,41)(36,44)(38,46)(39,43)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=g,e^2=h,f^2=h,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, 8679689);