#7649093

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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4968 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,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, 7556289);