#8093609

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	#34636	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 56 points:

(2,5)(6,8)(9,10,15,18)(11,13,17,14)(12,20,23,16)(19,22,24,21)(25,26,30,31)(27,35,36,32)(28,33,39,43)(29,41,42,34)(37,47,49,44)(38,45,50,48)(40,53,52,46)(51,54,56,55);(2,5)(3,7)(9,11,15,17)(10,14,18,13)(12,19,23,24)(16,22,20,21)(26,31)(27,36)(28,39)(34,41)(38,50)(40,52)(44,47)(54,55);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,47)(38,48)(39,43)(40,46)(41,42)(44,49)(45,50)(51,55)(52,53)(54,56);(1,3)(2,6)(4,7)(5,8)(9,12,15,23)(10,16,18,20)(11,19,17,24)(13,21,14,22)(25,27,30,36)(26,32,31,35)(28,37,39,49)(29,38,42,50)(33,44,43,47)(34,45,41,48)(40,51,52,56)(46,54,53,55);(9,13,15,14)(10,17,18,11)(12,21,23,22)(16,24,20,19)(25,28)(26,33)(27,37)(29,40)(30,39)(31,43)(32,44)(34,46)(35,47)(36,49)(38,51)(41,53)(42,52)(45,54)(48,55)(50,56);(9,14,15,13)(10,11,18,17)(12,22,23,21)(16,19,20,24)(25,29,30,42)(26,34,31,41)(27,38,36,50)(28,40,39,52)(32,45,35,48)(33,46,43,53)(37,51,49,56)(44,54,47,55)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=g,e^2=hi,f^2=g,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, 8000805);