#253774

order	512
nilpotency class	3
isoclinism	isoclinic class of #253699
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	#2635	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

798 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)(18,23)(20,21)(24,26)(25,31)(27,29)(30,32);(1,3)(2,8)(4,11)(5,12)(6,9)(7,13)(10,15)(14,16)(18,24,26,23)(25,31,32,30);(1,4,7,8)(2,9,11,12)(3,10,13,16)(5,15,6,14)(17,18,22,26)(19,25,28,32)(20,24,21,23)(27,30,29,31);(17,19,22,28)(18,25,26,32)(20,29,21,27)(23,30,24,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=i,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,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, 160970);