#470738

order	512
nilpotency class	3
isoclinism	isoclinic class of #454664
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

753 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)(3,5)(4,6)(7,8)(9,10,15,18,13,19,14,25)(11,16,20,34,22,30,26,28)(12,17,27,33,23,32,24,37)(21,29,35,40,36,31,38,39);(1,3,4,7)(2,5,6,8)(11,20)(14,15)(16,28)(18,25)(21,35)(22,26)(24,27)(29,39)(30,34)(31,40)(33,37)(36,38);(9,11)(10,16)(12,21,23,36)(13,22)(14,26)(15,20)(17,31,32,29)(18,34)(19,30)(24,38,27,35)(25,28)(33,39,37,40);(2,6)(5,8)(9,12)(10,17)(11,21)(13,23)(14,24)(15,27)(16,29)(18,33)(19,32)(20,35)(22,36)(25,37)(26,38)(28,39)(30,31)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377934);