#205227

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1116 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fhi,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112423);