#7792179

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	#38357	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4491 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)(9,10,14,16)(11,15,17,12)(13,18,21,23)(19,22,24,20)(25,26,30,34)(27,33,35,31)(28,29,32,39)(36,37,40,38);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,16)(13,19)(14,17)(18,22)(20,23)(21,24)(25,27)(26,31)(28,36)(29,38)(30,35)(32,40)(33,34)(37,39);(10,16)(11,17)(13,21)(20,22)(27,35)(31,33)(36,40)(37,38);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,16)(13,19)(14,17)(18,22)(20,23)(21,24)(25,28)(26,29)(27,36)(30,32)(31,38)(33,37)(34,39)(35,40);(9,12,14,15)(10,17,16,11)(13,20,21,22)(18,24,23,19)(25,29)(26,32)(27,37)(28,34)(30,39)(31,36)(33,40)(35,38);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,27)(26,33)(28,36)(29,37)(30,35)(31,34)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=ghi,e^2=h,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, 7699375);