#8094825

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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4992 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002021);