#8622172

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

3865 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,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, 8529368);