#360822

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂×#382
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2280 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1254 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240849, #72240851, #72240853, …
maximal normal subgroup of	#72240849, #72240851, #72240853, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268018);