#7678332

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

4527 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=gh,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, 7585528);