#6628874

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628872
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

16770 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)(6,11)(9,20)(12,25)(13,24)(14,21)(16,22)(19,31)(23,27)(26,35)(28,32)(30,33)(34,36);(5,6)(7,12)(8,13)(9,21)(10,14)(11,20)(15,26)(16,23)(17,27)(18,28)(19,32)(22,25)(24,31)(29,36)(30,34)(33,35);(5,7,10,17)(6,12,14,27)(8,15,18,29)(9,22,20,16)(11,23,21,25)(13,26,28,36)(19,33,31,30)(24,34,32,35);(2,4)(5,8)(6,13)(7,15)(9,19)(10,18)(11,24)(12,26)(14,28)(16,30)(17,29)(20,31)(21,32)(22,33)(23,34)(25,35)(27,36);(8,18)(13,28)(15,29)(19,31)(24,32)(26,36)(30,33)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262958, #72263037, #72267972, …
maximal normal subgroup of	#72262958, #72263037, #72267972, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536070);