#278512

order	512
nilpotency class	3
isoclinism	isoclinic class of #278491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

568 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)(3,9)(4,13)(5,7)(6,12)(8,11)(10,15)(14,16)(17,18,21,26,20,25,22,24)(19,27,30,32,29,23,28,31);(1,3,4,12,5,10,11,16)(2,6,8,9,7,14,13,15)(17,19,20,29)(18,23,25,27)(21,30,22,28)(24,32,26,31);(2,7)(3,10)(8,13)(12,16)(18,24)(19,28)(21,22)(23,27)(25,26)(29,30);(1,3,11,16,5,10,4,12)(2,6,13,15,7,14,8,9)(18,25)(23,27)(24,26)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185708);