#8622211

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3873 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529407);