#273149

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2617	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1048 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)(7,15,18,11)(8,17)(9,16,20,13)(10,14)(12,19)(21,22)(23,30)(24,26)(25,29)(27,34)(28,33)(31,35)(32,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,12)(10,18)(13,19)(14,15)(17,20)(22,26)(23,31)(24,33)(27,30)(28,29)(35,36);(2,4)(5,8)(6,12)(7,16)(9,11)(10,19)(13,18)(14,17)(15,20);(5,9)(6,13)(7,17)(8,11)(10,20)(12,18)(14,16)(15,19)(21,23,25,31)(22,27,29,36)(24,34,33,32)(26,35,28,30)

Polycyclic presentation:

2^9|a^2=fhi,d^2=i,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, 180345);