#7745609

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	#42951	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4885 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi

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, 7652805);