#267972

order	512
nilpotency class	3
isoclinism	isoclinic class of #267947
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	C₂³×C₄	C₂⁴
prime radical	#61572	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1001 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)(3,9,11,6)(4,12)(5,8)(7,13)(10,15,16,14)(17,18,22,24,20,25,21,26)(19,27,30,31,29,23,28,32);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,20,29)(18,23,25,27)(21,28,22,30)(24,32,26,31);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(18,25)(23,27)(24,26)(31,32)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi,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, 175168);