#10328272

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

4539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 10235468);