#147925

order	512
nilpotency class	4
isoclinism	isoclinic class of #136735
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#509	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1012 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)(5,6,10,17,12,18,11,26)(7,14,21,31,22,29,27,32)(8,16,9,25,24,13,23,15)(19,34,20,30,35,28,36,33);(6,13)(7,19)(8,9)(10,11)(14,29)(15,26)(16,18)(17,25)(20,27)(21,36)(22,35)(23,24)(28,30)(33,34);(2,4)(5,7)(6,14)(8,19)(9,20)(10,21)(11,27)(12,22)(13,28)(15,33)(16,34)(17,31)(18,29)(23,36)(24,35)(25,30)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250132, #72250135, #72250141, …
maximal normal subgroup of	#72250132, #72250135, #72250141, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55121);