#395843

order	512
nilpotency class	3
isoclinism	isoclinic class of #394034
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1176 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)(3,8)(4,6)(5,7)(9,10,15,30,14,20,13,19)(11,16,23,33,24,18,22,32)(12,25,29,17,28,37,27,36)(21,38,26,31,40,34,39,35);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,26)(13,22)(14,24)(15,23)(17,34)(18,20)(19,32)(21,27)(25,31)(28,39)(29,40)(30,33)(35,37)(36,38);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21)(13,29)(14,28)(15,27)(16,31)(18,35)(19,37)(20,36)(22,26)(23,39)(24,40)(25,30)(32,34)(33,38);(9,11)(10,18)(12,21)(13,23)(14,24)(15,22)(16,20)(17,35)(19,32)(25,38)(26,27)(28,40)(29,39)(30,33)(31,36)(34,37)

Polycyclic presentation:

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

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, 303039);