#239254

order	512
nilpotency class	3
isoclinism	isoclinic class of #233184
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

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:

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

Polycyclic presentation:

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

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, 146450);