#7707447

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4815 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,23)(19,22,25,29)(20,27,28,24)(26,32,31,30);(2,5)(6,8)(17,19)(18,22)(20,26)(21,25)(23,29)(24,30)(27,32)(28,31);(9,11)(10,13)(12,16)(14,15)(18,23)(19,25)(20,28)(30,32);(11,16)(13,15)(19,25)(22,29)(26,31)(30,32);(10,14)(13,15)(17,20,21,28)(18,24,23,27)(19,26,25,31)(22,30,29,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,e^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, 7614643);