#8445274

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

4100 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=h,e^2=hi,f^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, 8352470);