#239729

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

894 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi,f^c=fh,g^c=gi

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, 146925);