#455815

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×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1081 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)(3,4)(5,6)(7,16)(8,21)(9,12)(10,26)(11,15)(13,18)(14,24)(17,30)(19,29)(20,33)(22,32)(23,27)(25,31)(28,35)(34,36);(5,7,10,19,12,18,11,20)(6,13,15,29,9,16,26,33)(8,22,24,34,25,35,27,17)(14,30,21,28,23,36,31,32);(5,8)(6,14)(7,17,18,34)(9,23)(10,27)(11,24)(12,25)(13,28,16,32)(15,31)(19,35,20,22)(21,26)(29,30,33,36);(2,4)(5,9,12,6)(7,16,18,13)(8,23,25,14)(10,26,11,15)(17,28,34,32)(19,33,20,29)(21,24,31,27)(22,36,35,30)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,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, 363011);