#474285

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2572
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 381481);