#324504

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 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)(3,7)(4,5)(6,8)(9,10,13,15)(11,14,12,16)(17,18,21,24)(19,26,28,23)(20,30,25,22)(27,29,31,32);(2,5)(6,7)(9,11,15,16,13,12,10,14)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,31)(30,32);(9,12,13,11)(10,14,15,16)(17,20,22,18,21,25,30,24)(19,27,29,23,28,31,32,26);(1,3,4,8)(2,6,5,7)(18,24)(19,28)(20,25)(29,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=f,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,d^c=dh,g^c=gi

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, 231700);