#250753

order	512
nilpotency class	3
isoclinism	isoclinic class of #250647
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1224 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=i,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,d^b=dh,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

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316966, #72316975, #72316979
maximal normal subgroup of	#72316966, #72316975, #72316979

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157949);