#7859297

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	#47796	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4792 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7766493);