#417574

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×#3020	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2347 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)(7,13)(8,21)(9,15)(10,27)(11,16)(12,17)(14,25)(18,30)(19,33)(20,29)(22,31)(23,36)(24,34)(26,32)(28,35);(1,2)(3,4)(5,7,12,9)(6,13,17,15)(8,22,24,35)(10,20,11,19)(14,30,32,36)(16,33,27,29)(18,25,23,26)(21,28,34,31);(5,8)(6,14)(7,18)(9,23)(10,26)(11,25)(12,24)(13,28)(15,31)(16,34)(17,32)(19,22)(20,35)(21,27)(29,36)(30,33);(2,4)(5,9,12,7)(6,15,17,13)(8,23,24,18)(10,19,11,20)(14,31,32,28)(16,29,27,33)(21,30,34,36)(22,25,35,26)

Polycyclic presentation:

2^9|b^2=i,d^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, 324770);