#456920

order	512
nilpotency class	3
isoclinism	isoclinic class of #456400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1230 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)(3,4)(5,6)(7,8)(9,10,15,29,13,18,14,19)(11,20,30,32,22,16,23,36)(12,24,28,17,26,35,27,34)(21,37,39,33,25,40,38,31);(6,7)(9,11,15,30,13,22,14,23)(10,16,29,36,18,20,19,32)(12,25,28,38,26,21,27,39)(17,33,35,40,34,31,24,37);(5,7,8,6)(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,31)(18,34)(19,35)(20,33)(22,25)(23,39)(24,29)(30,38)(32,37)(36,40);(2,4)(6,7)(12,26)(17,34)(21,25)(24,35)(27,28)(31,33)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=de,f^c=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, 364116);