#8436215

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=g,f^2=g,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, 8343411);