#208529

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
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

801 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)(3,6)(4,5)(7,8)(10,14)(11,17)(12,20)(15,23)(16,22)(18,21)(25,26,30,33,29,32,28,38)(27,34,37,39,35,40,36,31);(9,10,12,14,13,16,20,22)(11,18,19,23,17,15,24,21)(25,27)(26,31,32,39)(28,36)(29,35)(30,37)(33,34,38,40);(2,5)(6,8)(9,11)(10,15)(12,19)(13,17)(14,21)(16,18)(20,24)(22,23)(26,32)(27,35)(33,38)(36,37);(1,3,4,7)(2,6,5,8)(10,16)(11,17)(14,22)(19,24)(27,35)(31,39)(34,40)(36,37)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,d^b=dh,d^c=dhi

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, 115725);