#289256

order	512
nilpotency class	3
isoclinism	isoclinic class of #289251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10176	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

617 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)(3,8)(4,14)(5,15)(6,10)(7,13)(9,11)(12,16)(17,18,20,24,21,25,28,23)(19,22,27,29,26,30,32,31);(1,3,11,4,6,12,5,13)(2,7,15,16,10,14,9,8)(17,19,20,27,21,26,28,32)(18,22,24,29,25,30,23,31);(3,12)(4,13)(7,14)(8,16)(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(1,4,11,12,6,13,5,3)(2,8,15,7,10,16,9,14)

Polycyclic presentation:

2^9|a^2=f,b^2=efh,d^2=eh,e^2=h,f^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

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, 196452);