#7714524

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

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

5636 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(21,29)(25,30)(26,32)(28,31);(1,3)(2,5)(4,8)(6,7)(17,18)(19,20)(21,25)(22,23)(24,27)(26,28)(29,30)(31,32);(2,6)(5,7)(9,10,12,14)(11,13,16,15)(18,23)(20,24)(25,30)(28,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,24)(18,19,23,27)(21,28,29,31)(25,26,30,32);(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)

Polycyclic presentation:

2^9|c^2=gh,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249417, #72249484, #72249491
maximal normal subgroup of	#72249417, #72249484, #72249491

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621720);