#244672

order	512
nilpotency class	3
isoclinism	isoclinic class of #244655
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

1135 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)(3,8)(4,14)(5,15)(6,10)(7,13)(9,12)(11,16)(17,18,20,25)(19,27,29,23)(21,24,22,26)(28,32,30,31);(1,3,6,11)(2,7,10,14)(4,12,13,5)(8,9,16,15)(17,19)(18,23)(20,29)(21,30)(22,28)(24,32)(25,27)(26,31);(3,11)(5,12)(7,14)(9,15)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32);(1,4)(2,8)(3,12)(5,11)(6,13)(7,9)(10,16)(14,15)(18,25)(23,27)(24,26)(31,32)

Polycyclic presentation:

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

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, 151868);