#7714560

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

5654 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,25)(21,23)(22,29)(26,30)(27,32)(28,31);(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,23)(21,25)(22,27)(26,31)(28,30)(29,32);(2,6)(3,7)(10,14)(13,15)(18,21)(19,20)(26,29)(27,28);(1,3,5,7)(2,4,6,8)(9,10)(11,13)(12,14)(15,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31);(1,4)(2,7)(3,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31);(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)

Polycyclic presentation:

2^9|d^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	#72249429, #72249484, #72249491
maximal normal subgroup of	#72249429, #72249484, #72249491

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621756);