#253569

order	512
nilpotency class	3
isoclinism	isoclinic class of #253507
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	#50715	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

892 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160765);