#119401

order	512
nilpotency class	3
isoclinism	isoclinic class of #100051
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=ei,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26597);