#242602

order	512
nilpotency class	3
isoclinism	isoclinic class of #233301
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

716 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=dh,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, 149798);