#432158

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1387 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)(5,6)(7,11,16,24)(8,12)(9,14)(10,13)(15,29,30,23)(17,25,31,32)(18,33,34,26)(19,28)(20,27)(21,22)(35,36);(5,7)(6,11,13,24)(8,18,19,34)(9,17)(10,16)(12,26)(14,25,21,32)(15,20,30,35)(22,31)(23,36)(27,29)(28,33);(5,8,10,19)(6,12,13,28)(7,15)(9,20,22,35)(11,23)(14,27,21,36)(16,30)(17,18)(24,29)(25,26)(31,34)(32,33);(2,4)(5,6)(7,11,16,24)(8,12)(9,21)(10,13)(14,22)(15,23,30,29)(17,32,31,25)(18,33,34,26)(19,28)(20,36)(27,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339354);