#7787754

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	#43919	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4295 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,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, 7694950);