#253269

order	512
nilpotency class	3
isoclinism	isoclinic class of #253267
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
prime radical	#40540	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1050 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160465);