#8069704

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

4916 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)(10,13)(14,16)(17,18,22,24)(19,20,25,23)(21,26,29,31)(27,28,32,30);(1,3)(2,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,25)(18,23,24,20)(21,27,29,32)(26,30,31,28);(1,2)(3,8)(4,6)(5,7)(18,24)(19,25)(21,29)(28,30);(1,4,5,8)(2,6,7,3)(9,11,12,15)(10,14,13,16)(17,20,22,23)(18,25,24,19)(21,30,29,28)(26,27,31,32);(1,4)(2,3)(5,8)(6,7)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,32)(29,31);(1,2)(3,4)(5,7)(6,8)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976900);