#6442564

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437445
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306292, #72311438, #72311454, …
maximal normal subgroup of	#72306292, #72311438, #72311454, …
maximal quotient of	#72306316, #72306320, #72306321

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349760);