#7641789

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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	#34633	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4876 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)(3,6,7,8)(10,14)(11,12)(17,18)(19,22)(20,28)(21,27)(23,25)(24,30)(26,29)(31,32);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,21)(20,26)(22,25)(23,27)(24,29)(28,31)(30,32);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,23,30)(18,24,25,28)(19,26,27,32)(21,29,22,31);(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,19,25,27)(20,29,30,31)(24,26,28,32);(20,30)(24,28)(26,32)(29,31);(17,22,23,21)(18,19,25,27)(20,31,30,29)(24,26,28,32)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi

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, 7548985);