#253608

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

939 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=hi,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, 160804);