#8463200

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

3902 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)(3,12,14,9)(4,15,10,5)(7,11,16,13)(18,24)(20,28)(21,25)(27,30)(35,41)(39,42)(43,47)(46,48);(1,3,8,14)(2,9,6,12)(4,7,10,16)(5,11,15,13)(17,18,23,24)(19,25,22,21)(20,26,28,31)(27,32,30,29)(33,34,38,37)(35,42,41,39)(36,44,45,40)(43,46,47,48);(1,4)(2,5)(3,7)(6,15)(8,10)(9,11)(12,13)(14,16)(17,19,23,22)(18,25,24,21)(20,27,28,30)(26,32,31,29)(33,35,38,41)(34,39,37,42)(36,43,45,47)(40,46,44,48);(1,5,8,15)(2,10,6,4)(3,13,14,11)(7,12,16,9)(17,20)(18,26)(19,27)(21,29)(22,30)(23,28)(24,31)(25,32)(33,36,38,45)(34,40,37,44)(35,43,41,47)(39,46,42,48);(1,6,8,2)(3,12,14,9)(4,15,10,5)(7,13,16,11)(17,21)(18,19)(20,29)(22,24)(23,25)(26,27)(28,32)(30,31);(1,7,8,16)(2,11,6,13)(3,10,14,4)(5,9,15,12)(17,22,23,19)(18,21,24,25)(20,27,28,30)(26,32,31,29)(33,37,38,34)(35,42,41,39)(36,40,45,44)(43,46,47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,e^2=hi,f^2=gi,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, 8370396);