#220316

order	512
nilpotency class	3
isoclinism	isoclinic class of #220244
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×#3341	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

856 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:

(1,2,6,10,7,3,4,8)(5,9,14,16,15,11,12,13)(17,18)(19,23)(20,28,30,24)(21,25)(22,26)(27,32,29,31);(1,3)(2,7)(4,10)(5,13)(6,8)(9,14)(11,12)(15,16)(17,19,22,21)(18,23,26,25)(20,27,30,29)(24,31,28,32);(1,4,7,6)(2,8,3,10)(5,12,15,14)(9,13,11,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(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=i,c^2=h,d^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,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, 127512);