#253239

order	512
nilpotency class	3
isoclinism	isoclinic class of #253219
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	#2617	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

908 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18)(19,25)(20,29)(21,24)(22,28)(23,31)(26,30)(27,32);(2,6)(7,15)(8,13)(9,16)(17,18)(19,26,27,22)(20,23)(21,24)(25,28,32,30)(29,31);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,19,21,27)(18,22,24,26)(20,30,31,28)(23,32,29,25);(2,6)(7,15)(8,13)(9,16)(20,31)(23,29)(25,32)(28,30)

Polycyclic presentation:

2^9|b^2=ehi,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,g^b=gh,e^c=eh,f^c=fi,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, 160435);