#242973

order	512
nilpotency class	3
isoclinism	isoclinic class of #242971
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238928, #72238930, #72238948
maximal normal subgroup of	#72238928, #72238930, #72238948

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150169);