#242522

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
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

724 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)(21,22,27,30)(23,31,34,36)(24,33,28,32)(25,29,26,35);(1,2)(3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,13)(14,19)(15,20)(21,23,26,33,27,34,25,32)(22,28,29,31,30,24,35,36);(2,4)(5,7)(6,12)(8,16)(9,18)(10,17)(11,13)(14,20)(15,19)(21,24,27,28)(22,23,30,34)(25,36,26,31)(29,32,35,33);(5,8,10,9)(6,13,15,14)(7,16,17,18)(11,19,20,12)(21,25,27,26)(22,29,30,35)(23,32,34,33)(24,36,28,31)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=h,d^2=i,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,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149718);