#219913

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127109);