#451898

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

769 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359094);