#6384010

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	D₄×#135	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2549 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)(5,6,10,16,11,17,29,15)(7,18,22,31,23,35,21,12)(8,24,27,36,20,13,26,30)(9,28,19,32,25,14,33,34);(5,7)(6,12)(8,25)(9,20)(10,22)(11,23)(13,32)(14,30)(15,35)(16,18)(17,31)(19,26)(21,29)(24,34)(27,33)(28,36);(2,4);(5,8)(6,13)(7,19)(9,21)(10,27)(11,20)(12,14)(15,36)(16,30)(17,24)(18,34)(22,25)(23,33)(26,29)(28,31)(32,35);(5,9,10,19,11,25,29,33)(6,14,16,34,17,28,15,32)(7,20,22,26,23,8,21,27)(12,30,18,24,31,36,35,13)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2957, #72343918
maximal normal subgroup of	D₄×#2957, #72343918
maximal quotient of	#72343934, #72343939

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291206);