#262976

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

833 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170172);