#7720004

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	#39130	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=i,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, 7627200);