#147676

order	512
nilpotency class	4
isoclinism	isoclinic class of #13404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,d^b=dh,f^b=fi,h^b=hi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239852, #72239853, #72239866, …
maximal normal subgroup of	#72239852, #72239853, #72239866, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54872);