#8104721

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279724, #72280742, #72281380, …
maximal normal subgroup of	#72279724, #72280742, #72281380, …
maximal quotient of	#72280171, #72281601, #72281613

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8011917);