#318564

order	512
nilpotency class	3
isoclinism	isoclinic class of #318549
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

541 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225760);