#461204

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

890 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)(5,6,10,24,11,13,9,15)(7,16,19,27,20,25,17,12)(8,14,22,29,23,28,21,30)(18,31,33,36,34,35,32,26);(5,7)(6,12,13,27)(8,18)(9,19)(10,17)(11,20)(14,26,28,36)(15,16,24,25)(21,33)(22,32)(23,34)(29,35,30,31);(6,13)(7,17)(8,21)(9,10)(12,25)(14,29)(16,27)(19,20)(22,23)(28,30)(31,35)(32,33);(2,4)(5,8,10,22,11,23,9,21)(6,14,24,29,13,28,15,30)(7,18,17,32,20,34,19,33)(12,26,25,35,27,36,16,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=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, 368400);