#7654564

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4854 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:

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

Polycyclic presentation:

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

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, 7561760);