#395021

order	512
nilpotency class	3
isoclinism	isoclinic class of #394020
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	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1541 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)(11,18)(12,24)(13,19)(14,29)(15,20)(16,22)(17,27)(21,34)(23,32)(25,35)(26,37)(28,36)(30,33)(31,39)(38,40);(2,5)(6,8)(9,11)(10,16)(12,25)(13,22)(14,23)(15,30)(17,34)(18,19)(20,32)(21,28)(24,31)(26,39)(27,38)(29,33)(35,37)(36,40);(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,31)(18,35)(19,36)(20,37)(22,38)(23,39)(24,29)(25,30)(32,34)(33,40);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,21)(13,22)(14,30)(15,23)(16,19)(17,35)(20,32)(24,40)(25,28)(26,38)(27,39)(29,33)(31,36)(34,37)

Polycyclic presentation:

2^9|d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 302217);