#7638383

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	#39742	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5064 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=gh,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, 7545579);