#7709908

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	#38838	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4485 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)(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(25,26,30,32)(27,35,36,31)(28,33,38,40)(29,39,37,34);(1,2,5,6)(3,7,4,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,27,30,36)(26,31,32,35)(28,37,38,29)(33,39,40,34);(10,15)(12,17)(16,22)(20,23)(26,32)(27,36)(28,38)(34,39);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,28)(26,33)(27,37)(29,36)(30,38)(31,39)(32,40)(34,35);(1,4,5,3)(2,8,6,7)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,39)(38,40);(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,29,30,37)(26,34,32,39)(27,28,36,38)(31,33,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617104);