#222121

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

941 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,9,14,10,13,8,12)(7,15,17,20,16,11,18,19)(21,22)(23,27)(24,32)(25,29)(26,30)(28,31)(33,36)(34,35);(1,2)(3,4)(7,16)(11,15)(17,18)(19,20)(21,23,26,25)(22,27,30,29)(24,33,34,31)(28,32,36,35);(5,7)(6,11)(8,17)(9,18)(10,16)(12,19)(13,15)(14,20)(21,24,26,34)(22,28,30,36)(23,31,25,33)(27,35,29,32);(2,4)(5,8,10,9)(6,12,13,14)(7,17,16,18)(11,19,15,20)(21,23,26,25)(22,29,30,27)(24,31,34,33)(28,32,36,35)

Polycyclic presentation:

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

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, 129317);