#452734

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

895 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:

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

Polycyclic presentation:

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

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, 359930);