#209610

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1025 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:

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

Polycyclic presentation:

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

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, 116806);