#110980

order	512
nilpotency class	3
isoclinism	isoclinic class of #110956
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

561 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18176);