#8097325

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

4171 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=gh,e^2=gh,f^2=g,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, 8004521);