#8463199

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

3917 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)(3,12,15,9)(4,16,10,5)(7,11,14,13)(18,23)(20,28)(24,26)(27,29)(35,41)(39,42)(43,47)(46,48);(1,3)(2,9)(4,13)(5,7)(6,12)(8,15)(10,11)(14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,34,38,37)(35,42,41,39)(36,44,45,40)(43,46,47,48);(1,4)(2,5)(3,13)(6,16)(7,9)(8,10)(11,15)(12,14)(17,19,22,21)(18,24,23,26)(20,27,28,29)(25,31,30,32)(33,35,38,41)(34,39,37,42)(36,43,45,47)(40,46,44,48);(1,5,8,16)(2,10,6,4)(3,14,15,7)(9,13,12,11)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,36,38,45)(34,40,37,44)(35,43,41,47)(39,46,42,48);(1,6,8,2)(3,12,15,9)(4,16,10,5)(7,13,14,11)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32);(1,7,8,14)(2,11,6,13)(3,5,15,16)(4,9,10,12)(17,21,22,19)(18,26,23,24)(20,27,28,29)(25,31,30,32)(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=i,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, 8370395);