#251955

order	512
nilpotency class	3
isoclinism	isoclinic class of #251935
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#9980	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159151);