#7721715

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	#50005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4420 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=hi,e^2=gh,f^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7628911);