#8436715

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

3915 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)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(33,34,40,38)(35,44,45,41)(36,46,42,37)(39,43,47,48);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(33,35,40,45)(34,41,38,44)(36,43,42,48)(37,47,46,39);(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,36)(34,37)(35,43)(38,46)(39,44)(40,42)(41,47)(45,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,37,40,46)(34,42,38,36)(35,39,45,47)(41,43,44,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30)(33,38,40,34)(35,44,45,41)(36,46,42,37)(39,48,47,43);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(33,39)(34,43)(35,37)(36,44)(38,48)(40,47)(41,42)(45,46)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=g,e^2=h,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, 8343911);