#6379180

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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	C₂×#3584	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4535 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)(5,6,10,16,11,17,28,15)(7,18,21,8,13,23,20,12)(9,14,26,32,27,33,25,34)(19,35,24,22,30,31,36,29);(5,7)(6,12)(8,17)(9,24)(10,21)(11,13)(14,31)(15,23)(16,18)(19,25)(20,28)(22,34)(26,30)(27,36)(29,32)(33,35);(2,4)(5,8)(6,13)(7,17)(9,22)(10,23)(11,12)(14,30)(15,21)(16,20)(18,28)(19,33)(24,34)(25,35)(26,31)(27,29)(32,36);(5,9)(6,14)(7,19)(8,22)(10,26)(11,27)(12,29)(13,30)(15,34)(16,32)(17,33)(18,35)(20,36)(21,24)(23,31)(25,28);(7,13)(8,12)(18,23)(19,30)(20,21)(22,29)(24,36)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343918
maximal normal subgroup of	#72343918
maximal quotient of	#72343951

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286376);