#250615

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61079	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1259 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316965, #72316972, #72316979
maximal normal subgroup of	#72316965, #72316972, #72316979

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157811);