#7806632

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

5351 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,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

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713828);