#7633746

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	#37452	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5024 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,28,31)(20,25,21,24)(27,32,29,30);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,32)(26,31);(11,15)(13,16)(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32);(11,15)(13,16)(17,21,22,20)(18,25,26,24)(19,27,28,29)(23,30,31,32);(10,14)(13,16)(18,26)(23,31)(24,25)(30,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^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, 7540942);