#8701275

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

3883 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8608471);