#7919991

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

4567 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7827187);