#8092119

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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

5135 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)(4,6)(7,8)(11,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(3,7)(5,8)(9,10)(11,16)(12,14)(13,15)(18,22)(23,28)(24,29)(30,32);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(3,7)(5,8)(10,14)(13,16)(18,22)(19,25)(24,29)(26,31);(2,6)(5,8)(18,22)(23,28)(24,29)(30,32);(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)

Polycyclic presentation:

2^9|c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281045
maximal normal subgroup of	#72281045

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7999315);