#183353

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₄⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

798 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,g^2=i,b^a=be,c^a=ch,d^a=dh,d^b=di,e^b=eh,f^b=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90549);