#8445594

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

5703 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)(3,7)(4,8)(5,6)(11,15)(13,16)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30);(2,6)(3,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,15)(14,16);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352790);