#273076

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1023 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180272);