#253642

order	512
nilpotency class	3
isoclinism	isoclinic class of #253603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

754 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=i,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=ghi,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, 160838);