#7692420

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	#34732	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5131 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=gi,e^2=ghi,f^2=i,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, 7599616);