#347812

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
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	#29660	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1010 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)(6,10)(8,11)(12,16)(13,17)(15,18)(19,20)(21,22)(23,30)(24,28)(25,27)(26,31)(29,33)(32,35)(34,36);(5,6,11,17,9,13,8,10)(7,12,18,20,14,19,15,16)(21,23)(22,26)(24,31)(25,32)(27,34)(28,30)(29,35)(33,36);(5,7)(6,12)(8,15)(9,14)(10,16)(11,18)(13,19)(17,20)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36);(2,4)(7,14)(12,19)(15,18)(16,20)(23,31)(26,30)(32,36)(34,35)

Polycyclic presentation:

2^9|b^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=di,d^c=dh,f^c=fi

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, 255008);