#254522

order	512
nilpotency class	3
isoclinism	isoclinic class of #254515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#32899	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1589 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)(3,5)(4,6)(7,8)(9,10)(11,16)(12,17)(13,15)(14,25)(18,24)(19,29)(20,34)(21,27)(22,28)(23,37)(26,31)(30,35)(32,36)(33,40)(38,39);(1,3)(2,5)(4,7)(6,8)(10,15)(11,19,21,26)(16,29,27,31)(17,28)(18,25)(20,33,35,38)(30,39,34,40)(32,37);(5,8)(9,11)(10,16)(12,20)(13,21)(14,26)(15,27)(17,30)(18,29)(19,24)(22,35)(23,38)(25,31)(28,34)(32,39)(33,36)(37,40);(9,12,13,22)(10,17,15,28)(11,20,21,35)(14,23,24,36)(16,30,27,34)(18,32,25,37)(19,33,26,38)(29,39,31,40)

Polycyclic presentation:

2^9|b^2=eh,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,g^c=gh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255859, #72256141, #72256159, …
maximal normal subgroup of	#72255859, #72256141, #72256159, …
maximal quotient of	#72262587, #72262610, #72262777

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161718);