#7673611

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	#44143	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4656 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)(11,12)(13,16)(14,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32);(1,2)(3,7)(4,8)(5,6)(9,11,14,13)(10,12,15,16)(18,22)(23,25)(24,28)(30,31);(1,3,5,4)(2,7,6,8)(10,15)(12,16)(17,18,21,22)(19,25,27,23)(20,24,29,28)(26,31,32,30);(9,12,14,16)(10,11,15,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,3,5,4)(2,7,6,8)(9,13,14,11)(10,16,15,12)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,4,5,3)(2,8,6,7)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,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, 7580807);