#108253

order	512
nilpotency class	3
isoclinism	isoclinic class of #108238
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

817 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257565, #72316632
maximal normal subgroup of	#72257565, #72316632
maximal quotient of	#72257863

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15449);