#6369745

order	512
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7582 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317304, #72317311, #72317322, …
maximal normal subgroup of	#72317304, #72317311, #72317322, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276941);