#8323781

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

4026 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8230977);