#417573

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3019	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2465 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:

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

Polycyclic presentation:

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

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, 324769);