#314209

order	512
nilpotency class	3
isoclinism	isoclinic class of #264363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

663 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=ei,d^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,g^b=gi,g^c=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, 221405);