#209789

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

925 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116985);