#220581

order	512
nilpotency class	3
isoclinism	isoclinic class of #220564
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	#3494	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

837 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)(3,7)(4,12,14,8)(5,9)(6,10)(11,16,13,15)(17,18,24,29)(19,30,25,32)(20,26,21,31)(22,28,23,27);(1,3,6,5)(2,7,10,9)(4,11,14,13)(8,15,12,16)(17,19)(18,20)(21,29)(22,26)(23,31)(24,25)(27,32)(28,30);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,20,24,21)(18,25,29,19)(22,30,23,32)(26,28,31,27);(1,5,6,3)(2,9,10,7)(4,13,14,11)(8,16,12,15)(17,21,24,20)(18,26,29,31)(19,27,25,28)(22,30,23,32)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127777);