#392025

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de

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, 299221);