#107535

order	512
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,10,16,8)(3,11,12,15,13,14,7,4)(17,18)(19,28,31,24)(20,32)(21,35)(22,27)(23,26)(25,34)(29,44,45,48)(30,39,36,40)(33,41)(37,42)(38,46,43,47);(1,3,5,12,6,13,16,7)(2,4,9,11,10,15,8,14)(17,19,22,30,23,31,21,36)(18,24,27,40,26,28,35,39)(20,29,34,47,33,45,37,46)(25,38,41,48,42,43,32,44);(1,4)(2,7)(3,9)(5,11)(6,15)(8,13)(10,12)(14,16)(17,20)(18,25)(19,29)(21,37)(22,34)(23,33)(24,38)(26,42)(27,41)(28,43)(30,47)(31,45)(32,35)(36,46)(39,44)(40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14731);