#460382

order	512
nilpotency class	3
isoclinism	isoclinic class of #454399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60453	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1312 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)(10,16)(12,26)(13,14)(17,21)(18,34)(19,20)(22,38)(23,24)(27,39)(28,29)(31,40)(32,33);(1,3,5,8)(2,6,4,7)(9,10,13,16,15,20,14,19)(11,21,23,17,25,32,24,33)(12,18,26,30,28,34,29,36)(22,37,27,31,39,35,38,40);(2,4)(6,7)(9,11)(10,17)(12,27)(13,24)(14,23)(15,25)(16,21)(18,35)(19,32)(20,33)(22,26)(28,38)(29,39)(30,31)(34,37)(36,40);(1,4,5,2)(3,7,8,6)(9,12,13,26,15,28,14,29)(10,18,16,30,20,34,19,36)(11,22,23,27,25,39,24,38)(17,31,32,35,33,40,21,37)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,d^2=e,e^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^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, 367578);