#460125

order	512
nilpotency class	3
isoclinism	isoclinic class of #455826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

1179 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^d=ei,f^d=fi,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, 367321);