#220331

order	512
nilpotency class	3
isoclinism	isoclinic class of #220264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3309	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

955 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127527);