#7658428

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

4979 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)(11,13)(12,15)(14,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30);(1,2)(3,7)(4,8)(5,6)(10,11)(14,15)(20,28)(24,31)(27,29)(30,32);(9,11,13,10)(12,15,16,14)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32);(1,3,5,4)(2,7,6,8)(9,12,13,16)(10,14,11,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32);(1,4,5,3)(2,8,6,7)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32);(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)

Polycyclic presentation:

2^9|c^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565624);