#456745

order	512
nilpotency class	3
isoclinism	isoclinic class of #454041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363941);