#7690347

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	#35927	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=gh,e^2=i,f^2=i,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, 7597543);