#111752

order	512
nilpotency class	3
isoclinism	isoclinic class of #99979
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 18948);