#7647008

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	#42584	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4614 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7554204);