#226925

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134121);