#6442727

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437621
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	C₂⁴	C₂⁵
prime radical	#62812	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4655 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)(7,11)(8,12)(9,13)(10,14)(15,22)(16,23)(17,24)(18,26)(19,27)(20,28)(21,29)(25,31)(30,34)(32,35)(33,36);(5,7,10,17)(6,11,14,24)(8,15,19,31)(9,20,21,16)(12,25,27,22)(13,28,29,23)(18,32,33,30)(26,34,36,35);(1,2,3,4)(5,8)(6,12)(7,15)(9,18)(10,19)(11,22)(13,26)(14,27)(16,30)(17,31)(20,32)(21,33)(23,34)(24,25)(28,35)(29,36);(5,8)(6,12)(7,15)(9,18)(10,19)(11,22)(13,26)(14,27)(16,30)(17,31)(20,32)(21,33)(23,34)(24,25)(28,35)(29,36);(5,9,10,21)(6,13,14,29)(7,16,17,20)(8,18,19,33)(11,23,24,28)(12,26,27,36)(15,30,31,32)(22,34,25,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306238, #72306242, #72306245, …
maximal normal subgroup of	#72306238, #72306242, #72306245, …
maximal quotient of	#72306461, #72306498

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349923);