#8425613

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4780 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)(3,7)(4,5)(6,8)(9,10,13,11)(12,15,16,14)(17,18,24,28)(19,29,31,25)(20,27,22,26)(21,30,32,23);(1,3)(2,6)(4,7)(5,8)(10,11)(14,15)(17,19,24,31)(18,25,28,29)(20,30,22,23)(21,26,32,27);(9,11,13,10)(12,15,16,14)(17,20,24,22)(18,26,28,27)(19,30,31,23)(21,25,32,29);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,14,11,15)(17,21,24,32)(18,23,28,30)(19,27,31,26)(20,25,22,29);(17,22,24,20)(18,27,28,26)(19,23,31,30)(21,29,32,25);(1,3)(2,7)(4,6)(5,8)(17,23)(18,21)(19,20)(22,31)(24,30)(25,26)(27,29)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332809);