#240312

order	512
nilpotency class	3
isoclinism	isoclinic class of #233080
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1016 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:

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

Polycyclic presentation:

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

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, 147508);