#7674480

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	#45034	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4675 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)(3,7)(4,8)(5,6)(10,14)(15,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(9,10,13,14)(11,15,12,16)(17,18,21,22)(19,23,25,29)(20,24,28,27)(26,30,31,32);(2,6)(7,8)(18,22)(19,25)(24,27)(26,31);(1,2)(3,7)(4,8)(5,6)(9,11,13,12)(10,15,14,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,16,14,15)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)

Polycyclic presentation:

2^9|b^2=g,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7581676);