#403975

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2573 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,h^b=hi,h^c=hi,e^d=ei,f^d=fi,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, 311171);