#205176

order	512
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1026 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)(7,12)(8,17)(9,19)(10,14)(11,15)(13,18)(16,20)(21,22,25,29,26,30,24,28)(23,31,32,27,34,35,33,36);(1,2,3,4)(5,7,9,15,10,16,18,8)(6,11,13,12,14,17,19,20)(21,23)(22,27)(24,32)(25,33)(26,34)(28,35)(29,31)(30,36);(5,8,18,16,10,15,9,7)(6,12,19,11,14,20,13,17);(21,24,26,25)(22,28,30,29)(23,32,34,33)(27,35,36,31)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112372);