#413379

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
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

902 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320575);