#402076

order	512
nilpotency class	3
isoclinism	isoclinic class of #396113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

792 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,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, 309272);