#7787755

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	#43899	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4331 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7694951);