#9723172

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

3797 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)(3,12)(4,9)(5,10)(6,8)(7,11)(13,16)(14,15)(17,18,23,21)(19,24,28,31)(20,29,30,25)(22,32,27,26)(36,44)(40,47)(43,45)(46,48);(1,3,7,6)(2,8,11,12)(4,13,5,14)(9,15,10,16)(17,19,23,28)(18,24,21,31)(20,27,30,22)(25,32,29,26)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(1,4,7,5)(2,9,11,10)(3,13,6,14)(8,15,12,16)(17,20,23,30)(18,25,21,29)(19,22,28,27)(24,26,31,32)(33,35,38,37)(34,39,41,42)(36,43,44,45)(40,46,47,48);(1,5,7,4)(2,10,11,9)(3,14,6,13)(8,16,12,15)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,32)(26,31)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,6,7,3)(2,8,11,12)(4,14,5,13)(9,15,10,16)(17,21,23,18)(19,24,28,31)(20,29,30,25)(22,26,27,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,2)(3,8)(4,9)(5,10)(6,12)(7,11)(13,15)(14,16)(17,22,23,27)(18,26,21,32)(19,20,28,30)(24,25,31,29)(33,37,38,35)(34,39,41,42)(36,45,44,43)(40,46,47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9630368);