#105541

order	512
nilpotency class	3
isoclinism	isoclinic class of #94972
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12737);