#110982

order	512
nilpotency class	3
isoclinism	isoclinic class of #110954
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18178);