#6615003

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37946	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257398, #72257437, #72264263
maximal normal subgroup of	#72257398, #72257437, #72264263
maximal quotient of	#72264751

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6522199);