#7633924

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,15,20,24)(12,18,13,16)(19,23,21,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(2,5)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,22)(17,24)(18,23)(25,27,30,37)(26,31,34,40)(28,36,29,35)(32,39,33,38);(9,12,14,13)(10,16,17,18)(11,19,20,21)(15,22,24,23)(25,28,30,29)(26,32,34,33)(27,35,37,36)(31,38,40,39);(10,17)(11,20)(16,18)(19,21)(25,29,30,28)(26,33,34,32)(27,36,37,35)(31,39,40,38);(9,13,14,12)(10,18,17,16)(11,21,20,19)(15,23,24,22)(26,34)(31,40)(32,33)(38,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 7541120);