#243277

order	512
nilpotency class	3
isoclinism	isoclinic class of #233265
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

582 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)(3,8)(4,6)(5,7)(9,10,14,16,13,17,12,18)(11,15,21,22,19,24,20,23)(25,26)(27,31)(28,37)(29,32)(30,33)(34,38)(35,39)(36,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,22)(17,24)(18,23)(25,27)(26,31)(28,35)(29,38)(30,36)(32,34)(33,40)(37,39);(1,4,5,3)(2,6,7,8)(10,16)(12,14)(15,22)(17,18)(20,21)(23,24)(25,26)(27,34)(28,32)(29,37)(30,33)(31,38)(35,40)(36,39);(10,17)(11,19)(16,18)(20,21)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,34,40,39)

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150473);