#405214

order	512
nilpotency class	3
isoclinism	isoclinic class of #393979
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2313 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)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,27)(22,30)(23,31)(25,35)(28,34)(29,33)(32,38)(36,37)(39,40);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,32)(18,21)(19,35)(20,31)(26,36)(27,37)(28,39)(29,38)(30,34)(33,40);(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,33)(19,26)(20,34)(21,36)(23,37)(24,39)(25,38)(31,32)(35,40);(2,6)(5,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,31,35)(17,33,34,26)(22,37,38,39)(30,36,32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285809, #72286082, #72286154
maximal normal subgroup of	#72285809, #72286082, #72286154

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312410);