#6629045

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629045
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2847
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72236859, #72240988, #72240989
maximal subgroup of	#72236859, #72236873, #72236877, …
maximal normal subgroup of	#72236859, #72236873, #72236890, …
maximal quotient of	#72251076, C₂×#6629045, #72255988, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536241);