#494957

order	512
nilpotency class	3
isoclinism	isoclinic class of #361285
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2290
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21107
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

533 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,d^b=dh,e^b=ei,f^c=fi,g^d=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, 402153);