#7804845

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

4917 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=gh,f^2=g,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, 7712041);