#112194

order	512
nilpotency class	3
isoclinism	isoclinic class of #112191
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

764 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19390);