#8104706

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

5052 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=i,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	#72280742
maximal normal subgroup of	#72280742
maximal quotient of	#72281414, #72281453

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8011902);