#455562

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

1127 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)(9,26)(10,14)(11,15)(12,18)(13,23)(17,34)(19,28)(20,33)(22,27)(24,30)(25,31)(29,35)(32,36);(5,7,9,19,11,18,10,20)(6,12,14,28,15,16,26,33)(8,22,23,17,25,36,24,35)(13,29,21,32,30,34,31,27);(5,8,11,25)(6,13,15,30)(7,17,18,35)(9,24,10,23)(12,27,16,32)(14,31,26,21)(19,22,20,36)(28,34,33,29);(2,4)(5,6)(7,12)(8,13)(9,14)(10,26)(11,15)(16,18)(17,32)(19,28)(20,33)(21,23)(22,29)(24,31)(25,30)(27,35)(34,36)

Polycyclic presentation:

2^9|b^2=f,c^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^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, 362758);