#7678293

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4521 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)(9,10,12,14)(11,13,16,15)(18,24)(22,26)(25,28)(30,32);(1,2)(3,7)(4,8)(5,6)(17,18,23,24)(19,28,20,25)(21,22,27,26)(29,30,31,32);(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,25,24,28)(21,31,27,29)(22,30,26,32);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,19)(18,25,24,28)(21,31,27,29)(22,32,26,30);(17,21,23,27)(18,26,24,22)(19,29,20,31)(25,30,28,32);(1,4,5,3)(2,8,6,7)(17,22,23,26)(18,27,24,21)(19,30,20,32)(25,29,28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585489);