#8093173

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	#35358	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5076 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)(9,10)(11,15)(12,13)(14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32);(2,5)(3,7)(10,12)(14,15)(18,22)(23,25)(24,28)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31);(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|c^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000369);