#365047

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272243);