#7630810

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

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

22282 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)(6,8)(14,28)(18,32)(23,37)(25,31)(26,38)(30,34)(35,39)(36,40);(5,6)(7,8)(9,10,15,19)(11,20,24,16)(12,13,17,27)(14,18,28,32)(21,33,29,22)(23,34,37,30)(25,26,31,38)(35,40,39,36);(1,2)(3,4)(5,6)(7,8)(9,11,15,24)(10,16,19,20)(12,21,17,29)(13,22,27,33)(14,23,28,37)(18,30,32,34)(25,35,31,39)(26,36,38,40);(9,12)(10,13)(11,21)(14,25)(15,17)(16,22)(18,26)(19,27)(20,33)(23,35)(24,29)(28,31)(30,36)(32,38)(34,40)(37,39);(9,13)(10,17)(11,22)(12,19)(14,26)(15,27)(16,29)(18,31)(20,21)(23,36)(24,33)(25,32)(28,38)(30,39)(34,35)(37,40);(9,14)(10,18)(11,23)(12,25)(13,26)(15,28)(16,30)(17,31)(19,32)(20,34)(21,35)(22,36)(24,37)(27,38)(29,39)(33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315572, #72315615
maximal normal subgroup of	#72315572, #72315615

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538006);