#7715828

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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^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, 7623024);