#226013

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=de,e^b=eh,e^d=eh

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, 133209);