#7799562

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	#44328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4467 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=gi,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, 7706758);