#7697321

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

4429 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)(5,8)(6,7)(9,10,12,14)(11,15,16,13)(17,18,23,26)(19,20,21,24)(22,25,30,32)(27,28,29,31);(1,3,6,5)(2,4,7,8)(9,11,12,16)(10,13,14,15)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32);(2,7)(4,8)(17,19,23,21)(18,24,26,20)(22,27,30,29)(25,31,32,28);(1,4,6,8)(2,3,7,5)(17,20,23,24)(18,19,26,21)(22,28,30,31)(25,27,32,29);(1,5,6,3)(2,8,7,4)(17,21,23,19)(18,24,26,20)(22,27,30,29)(25,28,32,31);(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7604517);