#445862

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15824
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

862 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:

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

Polycyclic presentation:

2^9|a^2=efg,c^2=fg,d^2=f,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,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^d=ei

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, 353058);