#112078

order	512
nilpotency class	3
isoclinism	isoclinic class of #99947
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=di,f^c=fh

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, 19274);