#239239

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

980 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fhi,g^c=ghi

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, 146435);