#8772734

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	#50590	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3910 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=gh,e^2=gh,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, 8679930);