#254475

order	512
nilpotency class	3
isoclinism	isoclinic class of #254467
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	#2404	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

949 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)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(18,22)(24,31);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,18)(19,25,27,32)(20,22)(21,24)(23,26,30,29)(28,31);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(17,19,20,27)(18,23,22,30)(21,29,28,26)(24,32,31,25);(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285077
maximal normal subgroup of	#72285077

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161671);