#109962

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

454 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17158);