#352393

order	512
nilpotency class	3
isoclinism	isoclinic class of #352384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2856
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

619 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=f,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=ci,d^b=di,d^c=di,g^d=gh

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, 259589);