#108044

order	512
nilpotency class	3
isoclinism	isoclinic class of #108026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ehi,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316741, #72316926, #72316929, …
maximal normal subgroup of	#72316741, #72316926, #72316929, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15240);