#7641992

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	#35292	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4814 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549188);