#7706620

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#50856	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5039 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,30)(29,32);(2,5)(6,8)(19,26)(20,28)(22,25)(23,30);(9,11,12,16)(10,13,14,15)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31);(11,16)(13,15)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31);(10,14)(13,15)(18,24)(19,26)(23,30)(27,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,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, 7613816);