#465712

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

899 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)(7,12)(8,20,22,34)(9,25)(10,15)(11,16)(13,18)(14,24,32,23)(17,30,21,31)(19,29)(26,28)(27,33,36,35);(1,2,3,4)(5,7)(6,12)(8,17)(9,18)(10,26)(11,19)(13,25)(14,27)(15,28)(16,29)(20,31)(21,22)(23,33)(24,35)(30,34)(32,36);(2,4)(5,7)(6,13)(8,21)(9,26)(10,18)(11,19)(12,25)(14,31)(15,29)(16,28)(17,22)(20,27)(23,33)(24,35)(30,32)(34,36);(5,8,9,23,11,22,10,24)(6,14,15,20,16,32,25,34)(7,17,18,33,19,21,26,35)(12,27,28,31,29,36,13,30)

Polycyclic presentation:

2^9|a^2=e,b^2=g,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,h^b=hi,d^c=de,f^c=fi,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, 372908);