#7690188

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4843 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7597384);