#250323

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	#61202	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,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, 157519);