#269771

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 176967);