#395035

order	512
nilpotency class	3
isoclinism	isoclinic class of #394098
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1555 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=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, 302231);