#262233

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

823 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)(7,15)(8,16)(9,12)(10,14)(11,18)(13,19)(17,20)(21,22,25,28)(23,26,30,35)(24,32,33,27)(29,36,31,34);(1,2,4,3)(5,7,9,16)(6,8,12,15)(10,11,19,17)(13,18,14,20)(21,23)(22,26)(24,29)(25,30)(27,34)(28,35)(31,33)(32,36);(5,8,9,15)(6,11,12,17)(7,14,16,13)(10,20,19,18)(23,29)(24,33)(26,34)(27,32)(30,31)(35,36);(1,3,4,2)(6,12)(8,15)(13,14)(18,20)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169429);