#7641763

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	#34743	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4859 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)(3,8)(4,6)(5,7)(10,14)(11,12)(17,18)(19,23)(20,28)(21,26)(22,25)(24,29)(27,30)(31,32);(1,3,5,4)(2,6,7,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,26)(18,23,25,21)(20,27,29,32)(24,31,28,30);(9,11,13,12)(10,15,14,16)(17,20,22,29)(18,24,25,28)(19,27,26,32)(21,30,23,31);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,15,14,16)(18,25)(19,26)(24,28)(27,32);(20,29)(24,28)(27,32)(30,31);(17,21,22,23)(18,26,25,19)(20,30,29,31)(24,32,28,27)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=g,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, 7548959);