#7985159

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×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892355);