#7689317

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
prime radical	#36213	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5771 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)(9,10)(11,14)(12,16)(13,15)(17,18,22,23)(19,24,27,29)(20,25,30,32)(21,26,31,28);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,14,15,12)(17,18,22,23)(19,24,27,29)(20,25,30,32)(21,26,31,28);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,14)(15,16)(18,23)(19,27)(25,32)(26,28);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,24)(20,28)(21,25)(22,27)(23,29)(26,30)(31,32);(17,20)(18,25)(19,28)(21,24)(22,30)(23,32)(26,27)(29,31);(17,21)(18,26)(19,25)(20,29)(22,31)(23,28)(24,30)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596513);