#456605

order	512
nilpotency class	3
isoclinism	isoclinic class of #453991
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

1380 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:

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

Polycyclic presentation:

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

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, 363801);