#267434

order	512
nilpotency class	3
isoclinism	isoclinic class of #267419
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	#3505	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

894 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)(3,11,14,7)(4,13)(5,12,16,9)(6,10)(8,15)(19,25)(22,31)(26,32)(27,30);(1,3,6,14)(2,7,10,11)(4,12,15,9)(5,13,16,8)(17,18,20,23)(19,26,25,32)(21,29,28,24)(22,30,31,27);(1,4)(2,8)(3,12)(5,7)(6,15)(9,14)(10,13)(11,16)(17,19)(18,22)(20,25)(21,30)(23,31)(24,32)(26,29)(27,28);(1,5)(2,9)(3,13)(4,7)(6,16)(8,14)(10,12)(11,15)(19,25)(22,31)(26,32)(27,30)

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,f^d=fh

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, 174630);