#120054

order	512
nilpotency class	3
isoclinism	isoclinic class of #99947
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

500 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27250);