#220541

order	512
nilpotency class	3
isoclinism	isoclinic class of #220528
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	#3488	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

934 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)(3,6)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18)(19,27)(20,24)(21,28)(22,26)(23,30)(25,32)(29,31);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19)(18,20)(21,29)(22,23)(24,32)(25,30)(26,31)(27,28);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,20,22,31)(18,23,26,19)(21,27,32,30)(24,25,29,28);(17,20,22,31)(18,24,26,29)(19,28,23,25)(21,30,32,27)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127737);