#7744964

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	#35777	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4897 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:

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

Polycyclic presentation:

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

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, 7652160);