#7648726

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4764 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7555922);