#7850124

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

4557 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7757320);