#8096135

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	#46080	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003331);