#7716079

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4366 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gh,e^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, 7623275);