#455819

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
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

1112 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)(6,11)(7,15)(9,20)(12,25)(13,23)(16,30)(17,21)(18,32)(22,27)(26,35)(28,36)(31,33);(5,6,9,14,10,11,20,24)(7,12,15,27,17,22,21,25)(8,13,18,29,19,23,32,34)(16,26,30,36,31,28,33,35);(5,7)(6,12,11,22)(8,16)(9,21)(10,17)(13,28,23,26)(14,25,24,27)(15,20)(18,33)(19,31)(29,36,34,35)(30,32);(2,4)(5,8,10,19)(6,13,11,23)(7,16,17,31)(9,18,20,32)(12,26,22,28)(14,29,24,34)(15,30,21,33)(25,35,27,36)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 363015);