#7804839

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

5647 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)(3,8)(4,6)(5,7)(17,18,22,21)(19,24,26,23)(20,28,30,25)(27,31,29,32);(1,3)(2,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,26)(18,23,21,24)(20,29,30,27)(25,32,28,31);(3,4)(6,8)(10,14)(12,15)(18,21)(19,26)(25,28)(27,29);(1,2)(3,8)(4,6)(5,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,32)(28,31)(29,30);(1,2)(3,8)(4,6)(5,7)(9,11)(10,12)(13,16)(14,15)(17,20,22,30)(18,25,21,28)(19,27,26,29)(23,31,24,32);(1,4,5,3)(2,6,7,8)(9,12)(10,11)(13,15)(14,16)(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=h,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, 7712035);