#114955

order	512
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1148 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22151);