#7630589

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4912 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)(3,8)(10,13)(14,16)(17,18)(19,28)(20,26)(21,24)(22,25)(23,27)(29,31)(30,32);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,24,27,28)(20,30,22,29)(25,32,26,31);(1,3)(2,7)(4,6)(5,8)(17,20,23,22)(18,25,27,26)(19,29,21,30)(24,31,28,32);(17,21,23,19)(18,24,27,28)(20,29,22,30)(25,32,26,31);(17,18)(19,28)(20,25)(21,24)(22,26)(23,27)(29,32)(30,31);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,22,23,20)(18,26,27,25)(19,29,21,30)(24,31,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537785);