#272248

order	512
nilpotency class	3
isoclinism	isoclinic class of #272155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,13)(10,14)(17,18,23,31,22,20,21,25)(19,24,28,26,29,30,27,32);(1,3,7,5)(2,8,11,10)(4,13,6,14)(9,15,16,12)(17,19)(18,24)(20,30)(21,27)(22,29)(23,28)(25,32)(26,31);(1,4,7,6)(2,9,11,16)(3,13,5,14)(8,15,10,12)(17,20,22,18)(19,26,29,32)(21,25,23,31)(24,27,30,28);(1,5,7,3)(2,10,11,8)(4,14,6,13)(9,12,16,15)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gi,f^c=fi,g^c=gi,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, 179444);