#352669

order	512
nilpotency class	3
isoclinism	isoclinic class of #352019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

764 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)(10,11)(12,13)(15,16)(18,22)(19,25)(20,28)(23,29)(24,26)(27,31);(2,5)(6,8)(9,10,13,15,14,16,12,11)(17,18,20,22,21,23,28,29)(19,26,25,32,27,24,31,30);(9,11,13,10,14,15,12,16)(18,23)(22,29)(24,26)(30,32);(1,3,4,7)(2,6,5,8)(9,10,12,11,14,16,13,15)(17,19,20,25,21,27,28,31)(18,24,22,30,23,26,29,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259865);