#360820

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

1274 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)(8,20,22,27)(9,14)(10,15)(11,18)(12,21,28,34)(13,23)(17,24,31,35)(19,25)(26,32)(29,30,33,36);(1,2)(3,4)(5,7,9,18,10,19,23,32)(6,11,13,16,15,26,14,25)(8,17,21,30,22,31,34,36)(12,24,27,29,28,35,20,33);(2,4)(5,8,10,22)(6,12)(7,17,19,31)(9,21,23,34)(11,24)(13,27)(14,20)(15,28)(16,29)(18,30,32,36)(25,33)(26,35);(5,7,9,18,10,19,23,32)(6,11,13,16,15,26,14,25)(8,17,21,30,22,31,34,36)(12,24,27,29,28,35,20,33)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=e,e^2=i,f^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	#72240850, #72240851, #72240852, …
maximal normal subgroup of	#72240850, #72240851, #72240852, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268016);