#8923660

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	#40346	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8830856);