#7696006

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

5827 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,25)(21,26)(22,24)(23,28)(29,31)(30,32);(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,29)(21,25)(22,28)(24,27)(26,30)(31,32);(2,5)(6,8)(18,24)(19,28)(25,32)(26,29);(1,2)(3,6)(4,5)(7,8)(17,19)(18,23)(20,29)(21,25)(22,28)(24,27)(26,30)(31,32);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,29)(21,23)(22,30)(24,32)(26,28)(27,31);(17,21)(18,26)(19,25)(20,27)(22,31)(23,30)(24,29)(28,32)

Polycyclic presentation:

2^9|a^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603202);