#8071412

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×#560	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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7978608);