#250876

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

1057 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=h,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, 158072);