#110997

order	512
nilpotency class	3
isoclinism	isoclinic class of #110956
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 18193);