#220333

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	C₂²×D₄	#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

822 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)(19,23)(20,28,30,24)(21,25)(22,26)(27,32,29,31);(1,3,6,2)(4,12,14,16)(5,9,7,11)(8,13,10,15)(17,19,22,21)(18,23,26,25)(20,27,30,29)(24,31,28,32);(2,3)(8,10)(9,11)(12,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(1,4)(2,8)(3,10)(5,15)(6,14)(7,13)(9,12)(11,16)(17,21,22,19)(18,25,26,23)(20,29,30,27)(24,32,28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,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, 127529);