#9843735

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	#43903	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4331 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9750931);