#7666335

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
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

6093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2)(3,6)(4,5)(7,8)(10,14)(15,16)(17,18)(19,21)(20,22)(23,24);(2,5)(6,8)(17,19,20,24)(18,21,22,23);(9,10)(11,16)(12,15)(13,14)(17,18)(19,23)(20,22)(21,24);(9,11,13,12)(10,15,14,16);(9,12,13,11)(10,15,14,16)(18,22)(21,23);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280765, #72304034
maximal normal subgroup of	#72280765, #72304034
maximal quotient of	#72281414, #72304075

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7573531);