#254357

order	512
nilpotency class	3
isoclinism	isoclinic class of #254347
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#18380	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=ehi,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,d^c=di,e^c=eh,f^c=fi,g^c=gi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161553);