#254533

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	#32890	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1567 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)(9,10,13,15)(11,16,21,27)(12,17,22,28)(14,25,24,18)(19,29,26,31)(20,34,35,30)(23,37,36,32)(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)(10,17)(11,20)(13,22)(14,23)(15,28)(16,30)(18,32)(19,33)(21,35)(24,36)(25,37)(26,38)(27,34)(29,39)(31,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,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	#72255929, #72256155, #72256159
maximal normal subgroup of	#72255929, #72256155, #72256159

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161729);