#443708

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,23)(15,25,27,20)(16,21,30,39)(18,24,19,34)(22,33,36,28)(26,41,42,35)(29,38,43,37)(31,40,32,44);(2,3)(5,7,8,11)(6,9,10,12)(13,15,17,27)(14,20,23,25)(16,29,30,43)(18,33,19,28)(21,37,39,38)(22,34,36,24)(26,32,42,31)(35,44,41,40);(6,10)(9,12)(13,16,19,32,17,30,18,31)(14,21,34,44,23,39,24,40)(15,26,33,43,27,42,28,29)(20,35,22,38,25,41,36,37);(1,3,4,2)(13,15)(14,22)(16,26)(17,27)(18,28)(19,33)(20,24)(21,38)(23,36)(25,34)(29,31)(30,42)(32,43)(35,40)(37,39)(41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 350904);