#404261

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311457);