#7633905

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
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	#37616	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4810 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)(11,14)(12,15)(13,16)(17,18,22,25)(19,23,28,32)(20,26,21,24)(27,31,29,30);(2,5)(6,8)(11,12)(14,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,32)(26,31);(9,11,13,12)(10,14,16,15)(17,20,22,21)(18,24,25,26)(19,27,28,29)(23,30,32,31);(9,12,13,11)(10,15,16,14)(18,25)(19,28)(24,26)(27,29);(10,16)(14,15)(17,21,22,20)(18,26,25,24)(19,29,28,27)(23,31,32,30);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 7541101);