#253816

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,d^2=h,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161012);