#8463188

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

3891 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8370384);