#348421

order	512
nilpotency class	3
isoclinism	isoclinic class of #348398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

932 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)(3,4)(5,6,9,12)(7,13,15,20)(8,17,14,11)(10,18,19,16)(21,22,26,31)(23,32,24,28)(25,30,27,35)(29,33,36,34);(5,7,14,18,9,15,8,16)(6,10,17,20,12,19,11,13)(21,23,26,24)(22,28,31,32)(25,34,27,33)(29,30,36,35);(21,24,26,23)(22,29,31,36)(25,34,27,33)(28,30,32,35);(2,4)(7,15)(10,19)(13,20)(16,18)(21,25,26,27)(22,30,31,35)(23,33,24,34)(28,29,32,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dhi,f^c=fi

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, 255617);