#253833

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

795 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:

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

Polycyclic presentation:

2^9|a^2=hi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161029);