#7712141

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	#35251	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5690 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30);(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(2,6)(3,7)(10,14)(13,15)(18,22)(19,25)(24,27)(26,31);(1,3,5,7)(2,4,6,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,4)(2,7)(3,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619337);