#209171

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

882 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116367);