#7711984

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619180);