#221131

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#61448	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

854 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=chi,d^b=de,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128327);