#441977

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 349173);