#7711129

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	#45788	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4464 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)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(25,26)(27,33)(28,31)(29,37)(30,32)(34,38)(35,39)(36,40);(1,2,5,6)(3,7,4,8)(25,27,30,28)(26,31,32,33)(29,35,38,36)(34,39,37,40);(10,15)(11,18)(17,20)(19,24)(26,32)(27,28)(34,37)(35,36);(1,3,5,4)(2,7,6,8)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,40)(37,38);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28,30,27)(26,33,32,31)(29,36,38,35)(34,40,37,39);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)

Polycyclic presentation:

2^9|b^2=g,d^2=gh,e^2=ghi,f^2=gh,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, 7618325);