#428563

order	512
nilpotency class	3
isoclinism	isoclinic class of #428398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2609
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 335759);