#7739319

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	#34976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4784 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)(3,4)(5,6)(7,8)(10,13)(14,16)(18,22)(19,25)(24,29)(27,31);(9,10)(11,14)(12,13)(15,16)(18,22)(20,28)(23,26)(27,31);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,3,5,7)(2,4,6,8);(1,4)(2,7)(3,6)(5,8)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)

Polycyclic presentation:

2^9|c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7646515);