#360364

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1049 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267560);