#7638338

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	#37817	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4812 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)(3,7)(4,6)(5,8)(17,18)(19,23)(20,27)(21,30)(22,25)(24,29)(26,32)(28,31);(1,3,5,4)(2,6,8,7)(10,13)(14,16)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,31,27,32);(1,3,5,4)(2,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,27)(30,32);(1,3,5,4)(2,7,8,6)(17,21,22,19)(18,23,25,30)(20,28,29,26)(24,31,27,32);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(20,29)(24,27)(26,28)(31,32);(1,4,5,3)(2,6,8,7)(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545534);