#7805050

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

5748 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:

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

Polycyclic presentation:

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

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, 7712246);