#273036

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
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	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1180 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)(7,13,9,11)(8,16)(10,14)(12,18)(15,20,17,19)(21,22)(23,30,32,36)(24,29)(25,27)(26,31,28,33)(34,35);(1,2)(3,4)(5,7,10,9)(6,11,14,13)(8,15,18,17)(12,19,16,20)(21,23)(22,26)(24,33)(25,32)(27,28)(29,36)(30,35)(31,34);(2,4)(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,20)(14,16)(22,27)(24,34)(26,28)(31,33);(5,9,10,7)(6,13,14,11)(8,17,18,15)(12,20,16,19)(21,23)(22,28)(24,33)(25,32)(26,27)(29,30)(31,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180232);