#455571

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1141 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:

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

Polycyclic presentation:

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

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, 362767);