#7631741

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×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4994 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538937);