#333024

order	512
nilpotency class	3
isoclinism	isoclinic class of #332983
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	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=i,c^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,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, 240220);