#8922974

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,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, 8830170);