#222866

order	512
nilpotency class	3
isoclinism	isoclinic class of #222720
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61498	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

872 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fhi,g^c=gh,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130062);