#8093619

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×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5306 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000815);