#384925

order	512
nilpotency class	3
isoclinism	isoclinic class of #31282
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

846 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)(7,15)(8,20)(9,24)(10,13)(11,18)(12,22)(14,25)(16,26)(17,28)(19,27)(21,29)(23,34)(30,35)(31,32)(33,36);(5,7)(6,11)(8,21)(9,17)(10,19)(12,29)(13,28)(14,27)(15,24)(16,23)(18,25)(20,31)(22,33)(26,30)(32,35)(34,36);(5,8,9,22,10,23,25,35)(6,12,14,20,13,30,24,34)(7,16,18,33,19,21,17,32)(11,26,15,31,28,29,27,36);(6,13)(7,17,19,18)(11,15,28,27)(12,30)(14,24)(16,32,21,33)(20,34)(26,31,29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292121);