#8772741

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

4062 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,16)(14,15)(17,18,23,28)(19,26,21,24)(20,30,29,25)(22,27,31,32)(35,41)(38,42)(43,47)(45,48);(2,7)(5,8)(9,15)(13,16)(17,19,23,21)(18,24,28,26)(20,31,29,22)(25,32,30,27)(33,34,39,36)(35,38,41,42)(37,44,46,40)(43,48,47,45);(1,3,6,11)(2,5,7,8)(4,12,14,10)(9,16,15,13)(17,20)(18,25)(19,22)(21,31)(23,29)(24,27)(26,32)(28,30)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3)(2,8)(4,10)(5,7)(6,11)(9,16)(12,14)(13,15)(17,21,23,19)(18,26,28,24)(20,31,29,22)(25,32,30,27)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,4)(2,9)(3,10)(5,13)(6,14)(7,15)(8,16)(11,12)(17,22,23,31)(18,27,28,32)(19,29,21,20)(24,30,26,25)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,5,6,8)(2,3,7,11)(4,13,14,16)(9,10,15,12)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,30)(31,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=h,e^2=h,f^2=gh,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, 8679937);