#8463202

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

3890 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:

(2,3)(4,11)(8,10)(12,16)(17,18,23,26)(19,27,30,22)(20,21,25,24)(28,31,32,29)(35,41)(39,42)(43,47)(46,48);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,23,30)(18,22,26,27)(20,28,25,32)(21,29,24,31)(33,34,38,37)(35,42,41,39)(36,44,45,40)(43,46,47,48);(17,20,23,25)(18,24,26,21)(19,28,30,32)(22,31,27,29)(33,35,38,41)(34,39,37,42)(36,43,45,47)(40,46,44,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,18)(19,27)(20,24)(21,25)(22,30)(23,26)(28,29)(31,32)(33,36,38,45)(34,40,37,44)(35,43,41,47)(39,46,42,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21,23,24)(18,25,26,20)(19,29,30,31)(22,32,27,28);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,27)(18,19,26,30)(20,31,25,29)(21,32,24,28)(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=h,d^2=gh,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, 8370398);