#346738

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	C₂×#3506	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=hi,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, 253934);