#7697271

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

4363 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=h,e^2=ghi,f^2=i,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, 7604467);