#7656513

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	#35821	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4848 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7563709);