#346101

order	512
nilpotency class	3
isoclinism	isoclinic class of #346096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

477 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=eh,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=defi,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253297);