#452274

order	512
nilpotency class	3
isoclinism	isoclinic class of #420055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2476
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16036
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

707 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:

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

Polycyclic presentation:

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

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, 359470);