#451045

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1281 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358241);