#347741

order	512
nilpotency class	3
isoclinism	isoclinic class of #347668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(6,10)(8,11)(12,17)(13,18)(14,20)(16,19)(21,22,25,28)(23,29,31,36)(24,32,34,27)(26,33,35,30);(5,6,11,18,9,13,8,10)(7,14,19,17,15,12,16,20);(5,7)(6,12)(8,16)(9,15)(10,17)(11,19)(13,14)(18,20)(21,23,25,31)(22,26,28,35)(24,33,34,30)(27,36,32,29);(2,4)(7,15)(12,14)(16,19)(17,20)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,c^b=ch,d^c=dh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254937);