#7657735

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=i,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564931);