#332953

order	512
nilpotency class	3
isoclinism	isoclinic class of #332911
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

646 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)(3,7)(4,12)(5,9)(6,10)(8,11)(13,15)(14,16)(17,18,21,26,23,27,22,32)(19,28,29,24,25,31,30,20);(1,3,6,5)(2,7,10,9)(4,11,13,14)(8,15,16,12)(17,18)(19,28)(20,29)(21,32)(22,26)(23,27)(24,30)(25,31);(1,4,6,13)(2,8,10,16)(3,11,5,14)(7,15,9,12)(17,19)(18,24)(20,27)(21,29)(22,30)(23,25)(26,31)(28,32);(2,9)(3,5)(7,10)(8,12)(11,14)(15,16)(17,20)(18,25)(19,27)(21,31)(22,28)(23,24)(26,29)(30,32)

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

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, 240149);