#253797

order	512
nilpotency class	3
isoclinism	isoclinic class of #253795
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	#35924	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

990 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)(8,22)(9,21)(10,13)(12,25)(14,26)(16,30)(20,23)(24,27)(28,31)(29,32);(2,8,9,23)(3,12)(4,15)(7,20,21,22)(10,16,24,32)(11,25)(13,29,27,30)(14,26)(17,18)(28,31);(1,2,6,10)(3,13,11,7)(4,16,18,8)(5,9,19,24)(12,27,25,21)(14,22,26,30)(15,23,17,32)(20,31,29,28);(3,12)(7,21)(11,25)(13,27)(14,28)(20,22)(26,31)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260639, #72260660
maximal normal subgroup of	#72260660

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160993);