#239712

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

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=di,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, 146908);