#8430431

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4245 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:

(4,12)(8,15)(10,16)(13,14)(18,21)(20,24)(25,29)(28,30)(33,34,40,38)(35,44,45,41)(36,46,42,37)(39,43,47,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,18,23,21)(19,24,27,20)(22,25,31,29)(26,30,32,28)(33,35,40,45)(34,41,38,44)(36,43,42,48)(37,47,46,39);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19,23,27)(18,24,21,20)(22,26,31,32)(25,30,29,28)(33,36)(34,37)(35,43)(38,46)(39,44)(40,42)(41,47)(45,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,19)(21,27)(22,30)(23,24)(25,32)(26,29)(28,31)(33,37,40,46)(34,42,38,36)(35,39,45,47)(41,43,44,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,23,18)(19,20,27,24)(22,29,31,25)(26,28,32,30)(33,38,40,34)(35,44,45,41)(36,46,42,37)(39,48,47,43);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,30)(27,32)(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=g,c^2=gh,d^2=h,e^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, 8337627);