#253317

order	512
nilpotency class	3
isoclinism	isoclinic class of #253315
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	#35436	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160513);