#7824438

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	#51581	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4526 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)(11,16)(12,20)(13,15)(14,18)(17,22)(19,23)(21,24)(25,26)(27,31)(28,36)(29,34)(30,32)(33,38)(35,40)(37,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,16)(17,23)(20,24)(21,22)(25,27,30,29)(26,31,32,34)(28,35,38,37)(33,39,36,40);(10,15)(11,18)(17,20)(19,21)(26,32)(31,34)(33,36)(39,40);(9,11)(10,16)(12,21)(13,18)(14,15)(17,23)(19,22)(20,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,38)(37,40);(1,3,5,4)(2,6,8,7)(9,12,13,22)(10,17,15,20)(11,19,18,21)(14,23,16,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,36)(34,40);(1,4,5,3)(2,7,8,6)(9,10)(11,16)(12,17)(13,15)(14,18)(19,24)(20,22)(21,23)(25,29,30,27)(26,34,32,31)(28,37,38,35)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7731634);