#455943

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1159 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)(3,4)(6,12)(7,16)(9,10)(13,17)(14,26)(18,32)(19,23)(20,24)(25,28)(27,33)(29,36)(34,35);(5,6,9,22,11,12,10,15)(7,17,16,25,19,28,23,13)(8,14,20,31,21,26,24,30)(18,33,32,29,35,36,34,27);(5,7)(6,13,12,25)(8,18)(9,23)(10,16)(11,19)(14,29,26,27)(15,17,22,28)(20,34)(21,35)(24,32)(30,36,31,33);(2,4)(5,8,11,21)(6,14,12,26)(7,18,19,35)(9,20,10,24)(13,27,25,29)(15,30,22,31)(16,32,23,34)(17,33,28,36)

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=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, 363139);