#7674464

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	#37845	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4692 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)(3,6)(4,8)(5,7)(10,13)(14,16)(17,18)(19,25)(20,22)(21,26)(23,24)(27,32)(28,29)(30,31);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,24)(19,20)(26,31)(27,28);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(17,19,23,20)(18,25,24,22)(21,28,30,27)(26,29,31,32);(1,2)(3,6)(4,8)(5,7)(17,20,23,19)(18,22,24,25)(21,28,30,27)(26,29,31,32);(17,21,23,30)(18,26,24,31)(19,27,20,28)(22,29,25,32);(1,4,5,3)(2,8,7,6)(17,22)(18,19)(20,24)(21,29)(23,25)(26,27)(28,31)(30,32)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,e^2=i,f^2=h,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, 7581660);