#360365

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,f^2=i,h^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, 267561);