#7683345

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	#45079	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4767 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)(7,8)(9,10,13,15)(11,12,14,16)(17,18,21,23)(19,22,25,30)(20,24,28,31)(26,29,27,32);(1,2)(3,7)(4,8)(5,6)(9,11,13,14)(10,12,15,16)(17,19)(18,22)(20,27)(21,25)(23,30)(24,32)(26,28)(29,31);(1,3,5,4)(2,7,6,8)(9,10)(11,12)(13,15)(14,16)(18,23)(19,25)(24,31)(26,27);(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,27)(21,25)(23,30)(24,32)(26,28)(29,31);(1,3,5,4)(2,7,6,8)(17,20,21,28)(18,24,23,31)(19,26,25,27)(22,29,30,32);(1,4,5,3)(2,8,6,7)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,32)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590541);