#346726

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2176 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)(5,6,10,13)(7,14,17,19)(8,12,9,18)(11,16,20,15)(22,27)(23,31)(24,25)(29,30)(32,33)(34,35);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,20)(14,18)(21,22,26,30)(23,28,32,36)(24,27,25,29)(31,34,33,35);(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,20,14)(21,23,26,32)(22,28,30,36)(24,31,25,33)(27,34,29,35);(2,4)(5,8,10,9)(6,12,13,18)(7,16,17,15)(11,14,20,19)(21,24,26,25)(22,29,30,27)(23,31,32,33)(28,35,36,34)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253922);