#111824

order	512
nilpotency class	3
isoclinism	isoclinic class of #10464
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

902 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)(5,6,9,12,10,7,8,11)(13,14,18,23,19,22,17,34)(15,24,28,39,26,20,27,36)(16,29,32,40,33,21,31,38)(25,41,43,44,42,35,30,37);(2,4)(5,7,10,6)(8,11,9,12)(13,15)(14,20)(16,30)(17,28)(18,27)(19,26)(21,37)(22,24)(23,39)(25,31)(29,44)(32,42)(33,43)(34,36)(35,38)(40,41);(13,16,19,33)(14,21,22,29)(15,25,26,42)(17,32,18,31)(20,35,24,41)(23,40,34,38)(27,43,28,30)(36,44,39,37)

Polycyclic presentation:

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

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, 19020);