#7638146

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	#52164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4828 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=gi,e^2=i,f^2=gh,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, 7545342);