#250282

order	512
nilpotency class	3
isoclinism	isoclinic class of #250199
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	#18882	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1077 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157478);