#8811212

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

4068 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)(17,18,22,26)(19,20,24,23)(21,25,30,32)(27,28,31,29)(33,34,39,37)(35,42,40,36)(38,45,46,41)(43,44,47,48);(1,3,7,8)(2,5,4,10)(6,14,15,11)(9,16,12,13)(17,19,22,24)(18,23,26,20)(21,27,30,31)(25,29,32,28)(34,37)(35,40)(41,45)(43,47);(2,4)(5,10)(9,12)(13,16)(17,20,22,23)(18,24,26,19)(21,29,30,28)(25,27,32,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,4,7,2)(3,10,8,5)(6,12,15,9)(11,16,14,13)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,36,39,42)(34,40,37,35)(38,44,46,48)(41,47,45,43);(1,5,7,10)(2,8,4,3)(6,13,15,16)(9,14,12,11)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37,39,34)(35,42,40,36)(38,45,46,41)(43,48,47,44);(1,6,7,15)(2,9,4,12)(3,11,8,14)(5,13,10,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=h,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, 8718408);