#7748714

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

5239 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)(4,5)(6,8)(10,14)(15,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,31)(29,32);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,29)(21,26)(23,25)(24,32)(27,30)(28,31);(2,5)(6,7)(18,23)(19,26)(24,28)(27,29);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,31)(28,32)(29,30);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,16,14,15)(17,20,21,30)(18,24,23,28)(19,27,26,29)(22,31,25,32);(17,18)(19,22)(20,24)(21,23)(25,26)(27,31)(28,30)(29,32)

Polycyclic presentation:

2^9|d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,e^b=ei,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, 7655910);