#7812077

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	#47828	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4459 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,f^2=g,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, 7719273);