#250810

order	512
nilpotency class	3
isoclinism	isoclinic class of #250775
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

988 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158006);