#6382745

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60643	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,d^2=f,f^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343622, #72343646, #72343648
maximal normal subgroup of	#72343622, #72343646, #72343648

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289941);