#7658084

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×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4667 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)(3,8)(4,6)(5,7)(10,13)(14,16)(17,18,23,24)(19,22,26,21)(20,28,31,25)(27,29,32,30);(9,10)(11,14)(12,13)(15,16)(18,24)(19,26)(25,28)(27,32);(1,3,5,4)(2,6,7,8)(17,19,23,26)(18,21,24,22)(20,27,31,32)(25,29,28,30);(17,20,23,31)(18,25,24,28)(19,27,26,32)(21,29,22,30);(1,4,5,3)(2,6,7,8)(17,21,23,22)(18,19,24,26)(20,29,31,30)(25,27,28,32);(1,2)(3,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,22,23,21)(18,26,24,19)(20,30,31,29)(25,32,28,27)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=h,f^2=g,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, 7565280);