#205225

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×#532	C₂²
LCS	G ▷ 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:

(2,4)(5,6)(7,16)(8,17)(9,14)(10,20)(11,15)(12,13)(18,19)(21,22,26,30,25,28,24,29)(23,31,32,27,34,36,33,35);(1,2,3,4)(5,7,10,13,11,17,9,19)(6,12,14,16,15,18,20,8)(21,23)(22,27)(24,32)(25,34)(26,33)(28,35)(29,36)(30,31);(5,8,10,12,11,16,9,18)(6,13,14,17,15,19,20,7)(22,28)(27,35)(29,30)(31,36);(5,9,11,10)(6,14,15,20)(7,13,17,19)(8,18,16,12)(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=eg,c^2=fi,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, 112421);