#6611343

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,20)(21,24)(22,27)(23,25)(26,28)(29,31)(30,32);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,20,18,23,27,22,25)(21,26,28,24,30,31,29,32);(1,3,4,8)(2,6,5,7)(11,16)(13,14)(17,20,23,22)(18,19,25,27)(21,28,30,29)(24,26,32,31);(9,11)(10,14)(12,16)(13,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,32)(27,31);(11,16)(13,14)(17,20,23,22)(18,19,25,27)(21,29,30,28)(24,31,32,26)

Polycyclic presentation:

2^9|b^2=fgh,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264053, #72264075, #72264263
maximal normal subgroup of	#72264053, #72264075, #72264263
maximal quotient of	#72308895

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6518539);