#253459

order	512
nilpotency class	3
isoclinism	isoclinic class of #253459
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	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

998 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)(9,12)(10,21)(13,27)(14,29)(16,17)(19,25)(20,26)(22,31)(24,32)(28,30);(2,8,10,25)(3,11)(4,16)(7,19,21,23)(9,24,26,31)(12,22,20,32)(13,27)(14,30)(15,17)(28,29);(1,2)(3,12)(4,8)(5,10)(6,9)(7,11)(13,20)(14,19)(15,31)(16,25)(17,24)(18,26)(21,27)(22,28)(23,29)(30,32);(2,9)(3,13)(7,20)(8,24)(10,26)(11,27)(12,21)(14,29)(19,32)(22,23)(25,31)(28,30)

Polycyclic presentation:

2^9|b^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fi,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, 160655);