#108294

order	512
nilpotency class	3
isoclinism	isoclinic class of #108282
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

743 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)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,21,26,23,28,22,27)(19,24,31,39,32,40,30,38)(20,33,35,41,34,42,36,25)(29,43,46,37,44,47,45,48);(2,6,9,8)(7,14,10,15)(17,19,22,30,23,32,21,31)(18,24,27,38,28,40,26,39)(20,29,35,46,34,44,36,45)(25,37,33,47,41,48,42,43);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)(17,20)(18,25)(19,29)(21,36)(22,35)(23,34)(24,37)(26,42)(27,33)(28,41)(30,46)(31,45)(32,44)(38,47)(39,43)(40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15490);