#7715734

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	#35003	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4556 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,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, 7622930);