#7697104

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

4582 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)(9,10,14,16)(11,15,13,12)(18,24)(20,26)(25,27)(28,30);(1,2,5,6)(3,7,4,8)(9,11,14,13)(10,15,16,12)(17,18,23,24)(19,27,21,25)(20,22,26,32)(28,31,30,29);(10,16)(12,15)(17,19,23,21)(18,25,24,27)(20,28,26,30)(22,29,32,31);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,16)(14,15)(17,20,23,26)(18,22,24,32)(19,28,21,30)(25,29,27,31);(9,13,14,11)(10,12,16,15)(17,21,23,19)(18,25,24,27)(20,30,26,28)(22,29,32,31);(1,4,5,3)(2,8,6,7)(17,22)(18,26)(19,29)(20,24)(21,31)(23,32)(25,30)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7604300);