#253605

order	512
nilpotency class	3
isoclinism	isoclinic class of #253603
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	#35684	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

914 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)(8,21)(9,20)(10,13)(12,25)(14,26)(16,29)(19,22)(23,27)(24,30)(28,31);(2,8,9,22)(3,12)(4,15)(7,19,20,21)(10,24,23,16)(11,25)(13,29,27,30)(14,31)(18,32)(26,28);(1,2,6,10)(3,13,11,7)(4,16,15,22)(5,9,17,23)(8,18,24,32)(12,27,25,20)(14,19,26,29)(21,31,30,28);(3,12)(7,20)(11,25)(13,27)(14,28)(19,21)(26,31)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233848
maximal normal subgroup of	#72233848

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160801);