#360971

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

736 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239855, #72239860, #72240851, …
maximal normal subgroup of	#72239855, #72239860, #72240851, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268167);