#7716224

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	#35133	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4299 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

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, 7623420);