#234935

order	512
nilpotency class	3
isoclinism	isoclinic class of #234934
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	#35088	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2273 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)(9,10,14,18)(11,19,22,33)(12,16,23,31)(13,25,27,17)(15,26,29,21)(20,35,36,34)(24,39,38,32)(28,37,30,40);(5,8)(9,11)(10,15)(12,20)(13,26)(14,22)(16,30)(17,33)(18,29)(19,25)(21,27)(23,36)(24,40)(28,31)(32,34)(35,39)(37,38);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,28)(17,32)(18,31)(19,34)(21,37)(22,36)(25,39)(26,40)(27,38)(29,30)(33,35);(12,23)(16,31)(20,36)(24,38)(28,30)(32,39)(34,35)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260824, #72260854
maximal normal subgroup of	#72260824, #72260854
maximal quotient of	#72260938

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142131);