#242540

order	512
nilpotency class	3
isoclinism	isoclinic class of #233184
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

728 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=hi,d^2=h,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, 149736);