#6468708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437412
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	#23923	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257267, #72257512, #72264009
maximal normal subgroup of	#72257267, #72257512, #72264009
maximal quotient of	#72264942

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375904);