#245441

order	512
nilpotency class	3
isoclinism	isoclinic class of #245375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×H(5,𝔽₂)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1061 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152637);