#206761

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^b=eh,g^b=gh,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, 113957);