#267112

order	512
nilpotency class	3
isoclinism	isoclinic class of #267051
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	C₂×#3106	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1030 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)(5,6,10,18,9,13,8,12)(7,14,15,11,17,19,16,20)(21,22,25,29,24,28,34,27)(23,30,31,36,32,26,33,35);(2,4)(21,23)(22,26)(24,32)(25,31)(27,36)(28,30)(29,35)(33,34);(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,20)(14,18)(22,27)(23,31)(25,34)(28,29)(32,33)(35,36);(2,4)(5,8,9,10)(6,12,13,18)(7,15,17,16)(11,19,20,14)(22,28)(26,30)(27,29)(35,36)

Polycyclic presentation:

2^9|a^2=fhi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174308);