#239070

order	512
nilpotency class	3
isoclinism	isoclinic class of #233152
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×#3169	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1052 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)(7,15)(8,11)(9,13)(10,14)(12,16)(17,20)(18,19)(21,22)(23,31)(24,28)(25,29)(26,30)(27,34)(32,35)(33,36);(6,11)(7,16)(8,9)(12,19)(13,14)(17,18)(21,22)(23,32)(24,28)(25,29)(26,30)(27,33)(31,35)(34,36);(2,4)(5,7,8,18,10,17,9,16)(6,12,13,20,14,19,11,15)(21,23)(22,27)(24,33)(25,34)(26,35)(28,31)(29,32)(30,36);(7,17)(12,19)(15,20)(16,18)(21,24,26,25)(22,28,30,29)(23,33,35,34)(27,31,36,32)

Polycyclic presentation:

2^9|c^2=e,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146266);