#242546

order	512
nilpotency class	3
isoclinism	isoclinic class of #233184
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

696 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:

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

Polycyclic presentation:

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

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, 149742);